\documentclass[12pt]{amsart} \setlength{\oddsidemargin}{0pt} \setlength{\evensidemargin}{0pt} \setlength{\textwidth}{17cm} \setlength{\textheight}{22cm} \usepackage{graphics} \usepackage{amsthm} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} \usepackage{epic} \usepackage{curves} \usepackage{epsfig} \usepackage{bm} \usepackage{amscd} \input{psfig.sty} \input diagrams \newcommand{\baseRing}[1]{\ensuremath{\mathbb{#1}}} \newcommand{\Z}{\baseRing{Z}} \newcommand{\C}{\baseRing{C}} \newcommand{\N}{\baseRing{N}} \newcommand{\R}{\baseRing{R}} \newcommand{\Q}{\baseRing{Q}} \newcommand{\PP}{\baseRing{P}} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{construction}[theorem]{Construction} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{observation}[theorem]{Observation} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \newtheorem{algorithm}[theorem]{Algorithm} \def\theenumi{\roman{enumi}} \renewcommand{\labelenumi}{\theenumi.} \def\Av{A\cdot (\eta + \alpha)} \def\supp{ {\rm supp}\, } \def\std{ {\rm Std}\, } \def\ass{ {\rm Ass}\, } \def\rank{ {\rm rank}\, } \def\vol{ {\rm vol}\, } \def\gl{ {\rm GL} } \def\conv{ {\rm conv} } \def\nsupp{ {\rm nsupp} } \def\lsupp{ {\rm lsupp} } \def\ini{ {\rm in}\, } \def\ind{ {\rm ind}\, } \def\ker{ {\rm ker}\, } \def\coker{ {\rm coker}\, } \def\re{ {\rm Re}\, } \def\Span{ {\rm Span}\, } \def\vert{ {\rm vert}\, } \def\im{ {\rm im}\, } \def\ord{ {\rm ord}\, } \def\symb{ {\rm symb}\, } \def\char{ {\rm char}\, } \def\sing{ {\rm sing}\, } \def\New{ {\rm New}\, } \def\Horn{ {\rm Horn}\, } \def\sign{ {\rm sign}\, } \def\codim{ {\rm codim}\, } \title{Bivariate hypergeometric $D$-modules} \author{Alicia Dickenstein} \address{Dto. de Matem\'atica, FCEyN, Universidad de Buenos Aires, \newline \indent (1428) Buenos Aires, Argentina.} \email{alidick@dm.uba.ar} \thanks{Alicia Dickenstein was partially supported by UBACYT X052 and ANPCYT 03-6568, Argentina.} \author{Laura Felicia Matusevich} \address{Department of Mathematics, Harvard University, \newline \indent Cambridge, MA 02138, USA.} \email{laura@math.harvard.edu} \thanks{Laura Felicia Matusevich was partially supported by a Sarah M. Hallam fellowship at UC Berkeley, and Liftoff fellowship from the Clay Mathematics Institute.} \author{Timur Sadykov} \address{Department of Mathematics, \newline \indent The University of Western Ontario, \newline \indent London, Ontario N6A 5B7, Canada.} \email{tsadykov@uwo.ca} \thanks{Timur Sadykov was partially supported by the Russian Ministry of Education, grant E~02-1-138.} \begin{document} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We undertake the study of bivariate Horn systems for generic parameters. We prove that these hypergeometric systems are regular holonomic, and we provide an explicit formula for their holonomic rank as well as bases of their spaces of complex holomorphic solutions. We also obtain analogous results for the generalized hypergeometric systems arising from lattices of any rank. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \maketitle \section{Introduction} \label{sec:intro} Classically, there have been two main directions in the study of hypergeometric functions. The first of these is to study the properties of a particular series, analyze its convergence, compute its values at some specific points providing combinatorial identities, give integral representations, and find relations with other series of the same kind. Here one could refer to well known works of Gauss and Euler, for instance,~\cite{euler} and~\cite{gauss}. The other classical avenue of research is to find a differential equation that our hypergeometric function satisfies, and to study all the solutions of that equation. This approach was pioneered by Kummer, who showed that the Gauss hypergeometric function: \begin{align*} f(z) & =F[a,b;c;z] \\ & = 1 + \frac{ab}{c}\frac{z}{1!}+ \frac{a(a+1)b(b+1)}{c(c+1)}\frac{z^2}{2!}+ \frac{a(a+1)(a+2)b(b+1)(b+2)}{c(c+1)(c+2)}\frac{z^3}{3!}+\cdots \end{align*} satisfies the differential equation: \[ z(1-z)\frac{d^2f}{d z^2}+(c-(1+a+b)z)\frac{d f}{d z}-abf=0. \] Kummer went on to find all of the solutions of this equation (see~\cite{kummer}). He constructed twenty-four (Gauss) series that, whenever~$a$,~$b$ and~$c$ are not integers, provide representations of two linearly independent solutions to the Gauss equation, that are valid in any region of the complex plane. Riemann also had a fundamental influence in this field \cite{riemann}. For more historical details on hypergeometric functions, and a comprehensive treatment of their classical theory, see \cite{slater}. Both of these approaches have been tried for bivariate hypergeometric series. In his article~\cite{erdelyi}, Erd\'elyi gives a complete set of solutions for the following system of two hypergeometric equations in two variables: \begin{align*} \big( x (\theta_x + \theta_y +a)(\theta_x+b)-\theta_x(\theta_x+\theta_y+c-1) \big) f = & \; 0 \; , \\ \big( y (\theta_x + \theta_y +a)(\theta_y+b')-\theta_y(\theta_x+\theta_y+c-1) \big) f = & \; 0 \; , \end{align*} where $\theta_x=x\frac{\partial}{\partial x}$ and $\theta_y=y \frac{\partial}{\partial y}$. This is the system of equations for Appell's function~$F_1$, and for generic values of the parameters~$a$,~$b$,~$b'$ and~$c$, Erd\'elyi constructs more than~$120$ fully supported series solutions through contour integration. By a {\em fully supported series}, we mean a series such that the convex hull of the exponents of the monomials appearing with nonzero coefficient contains a full dimensional cone. The holonomic rank of this system, that is, the dimension of its space of complex holomorphic solutions around a nonsingular point, is~$3$. Another interesting system of two second order hypergeometric equations in two variables is: \begin{align*} \big( x(2 \theta_x-\theta_y+a')(2\theta_x-\theta_y+a'+1)- (-\theta_x+2\theta_y+a)\theta_x \big) f = & \; 0 \; , \\ \big( y(-\theta_x+2\theta_y+a)(-\theta_x+2\theta_y+a+1)- (2\theta_x-\theta_y+a')\theta_y \big) f = & \; 0 \; . \end{align*} This is the system of equations for Horn's function~$G_3$, and its holonomic rank is~$4$. Erd\'elyi notes that, in a neighborhood of a given point, three linearly independent solutions of this system can be obtained through contour integral methods. He also finds a fourth linearly independent solution: the Puiseux monomial $x^{-(a+2a')/3} y^{-(2a+a')/3}$. He remarks that the existence of this elementary solution is puzzling, especially since it cannot be expressed using contour integration, and offers no explanation for its ocurrence. One of the goals of this article is to give a formula for the rank of a system of two hypergeometric equations in two variables when the parameters are generic. We will explain why the system for Appell's~$F_1$ has rank~$3$ and why the very similar system for Horn's~$G_3$ has rank~$4$. We will also show that Puiseux polynomial solutions are a commonplace phenomenon. Moreover, we will prove that these systems of hypergeometric equations are regular holonomic for a generic choice of the parameters. Our starting point are the ideas of Gel$'$fand, Graev and Retakh~\cite{gelfand-graev-retakh} about the $\Gamma$-series associated with lattices, and how they relate to Horn series. Notice that $\Gamma$-series as defined in~\cite{gelfand-graev-retakh} are fully supported, and they do not account for the Puiseux polynomial solutions of Horn systems. Although it has been stated in the literature, e.g.~\cite[p. 296]{GKZbook}, that Horn systems given by nonconfluent hypergeometric operators (the situation which we study) are regular holonomic, we are not aware of a published proof, even for the case of systems of equations of order two. Since we will be dealing with lattices that are not necessarily saturated, we also need to study the generalized hypergeometric systems associated with lattices (more general than the $A$-hypergeometric systems of Gel$'$fand, Kapranov and Zelevinsky). We prove that, for generic parameters, these systems are also regular holonomic, without restriction on the number of variables or rank of the corresponding lattice, and prove the expected formula for their generic holonomic rank. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Multivariate hypergeometric systems} \label{sec:formula} In order to accommodate two different sets of variables, we denote by~$D_n$ the Weyl algebra with generators $x_1,\dots, x_n$, $\partial_{x_1},\dots,\partial_{x_n}$, and by~$D_m$ the Weyl algebra whose generators are $y_1, \dots ,y_m$, $\partial_{y_1},\dots ,\partial_{y_m}$. We set $\theta_{x_j}=x_j \partial_{x_j}$ for $1 \leq j \leq n$, and $\theta_{y_i}=y_i\partial_{y_i}$, for $ 1 \leq i \leq m$. We also define $\theta_x=(\theta_{x_1}, \dots , \theta_{x_n})$ and $\theta_y=(\theta_{y_1},\dots ,\theta_{y_m})$. When the meaning is clear, we will drop many of the subindices to simplify the notation. We fix a matrix $A=(a_{ij}) \in \Z^{(n-m) \times n}$ of full rank~$n-m$ whose first row is the vector $(1,\dots ,1)$, and a matrix ${\mathcal{B}} \in \Z^{n \times m}=(b_{ji})$ of full rank~$m$ such that $A \cdot {\mathcal{B}} = 0$. For $1 \leq j \leq m$, set $b_j = (b_{j1},\dots , b_{jm}) \in \Z^m$ the~$j$-th row of~${\mathcal{B}}$. The (positive) greatest common divisor of the maximal minors of the matrix~${\mathcal{B}}$ is denoted by $g$. \smallskip %\pagebreak For $i = 1,\dots , m$, and a fixed parameter vector $c=(c_1,\dots, c_n) \in \C^n$, we let \begin{align}\label{def:horn} &{\bm{P}}_i = \prod_{b_{ji}<0} \prod_{l=0}^{|b_{ji}|-1}( b_j \cdot \theta_y + c_j-l), \\ &\bm{Q}_i = \prod_{b_{ji}>0} \prod_{l=0}^{b_{ji}-1} (b_j \cdot \theta_y + c_j-l) , \; \mbox{and} \\ & H_i = \bm{Q}_i-y_i\bm{P}_i , \end{align} where $ b_j \cdot \theta_y = \sum_{k=1}^m b_{jk} \theta_{y_k}.$ The operators~$H_i$ are the {\em Horn operators} corresponding to the lattice $L_{{\mathcal{B}}}=\{ {\mathcal{B}}\cdot z : z \in \Z^m \}$ and the parameter vector~$c$. We call $d_i=\sum_{b_{ij}>0} b_{ij} = - \sum_{b_{ij}<0} b_{ij}$ the order of the operator~$H_i$. \begin{definition} The {\em Horn system} is the following left ideal of~$D_m$: \[ \Horn({\mathcal{B}},c) = \langle H_1,\dots ,H_m \rangle \subseteq D_m .\] \end{definition} Now denote by~$b^{(i)}$ the columns of the matrix~$\mathcal{B}$. Any vector $u \in \R^n$ can be written as $u=u_+-u_-$, where $(u_+)_i = \max(u_i,0)$, and $(u_-)_i= - \min(u_i,0)$. For $i=1,\dots ,m$, we let: \[ T_i = \partial_x^{b^{(i)}_+} - \partial_x^{b^{(i)}_-}, \] here we use multi-index notation $\partial_x^v=\partial_{x_1}^{v_1}\cdots \partial_{x_n}^{v_n}$. More generally, for any $u \in L_{{\mathcal{B}}}$, set \[ T_u = \partial_x^{u_+}-\partial_x^{u_-} . \] These are the {\em lattice operators} arising from $L_{{\mathcal{B}}}$. \begin{definition} The {\em lattice ideal} arising from~$L_{{\mathcal{B}}}$ is: \[I_{{\mathcal{B}}} = \langle T_u : u \in L_{{\mathcal{B}}} \rangle \subseteq \C[\partial_{x_1}, \dots ,\partial_{x_n} ] .\] Recall that the {\em toric ideal} corresponding to~$A$ is: \[ I_A = \langle T_u : u \in \ker_{\Z}(A) \rangle \subseteq \C[\partial_{x_1}, \dots ,\partial_{x_n} ] .\] We will also denote: \[ I = \langle T_1,\dots ,T_m \rangle \subseteq \C[\partial_{x_1}, \dots ,\partial_{x_n} ] .\] The ideal~$I$ is called a {\em lattice basis ideal}. Notice that, for~$m=2$, $I$ is a complete intersection. This is not necessarily true if~$m>2$. \end{definition} Lattice ideals and toric ideals have been extensively studied (see, for instance,~\cite{binomialideals}, \cite{gb&cp}). Lattice basis ideals were introduced in \cite{latticebasis}. There is a natural system of differential equations arising from a toric ideal $I_A$ and a parameter vector. This system, called the {\em $A$-hypergeometric system with parameter~$A\cdot c$}, is defined as: \[ H_A(A\cdot c) = I_A + \langle \sum_{j=1}^n a_{ij}x_j \partial_{x_j} - (A\cdot c)_i : i=1,\dots ,n-m \rangle \subseteq D_n .\] >From now on we will use the notation $\langle A\cdot \theta - A\cdot c \rangle$ to mean $\langle \sum_{j=1}^n a_{ij}x_j \partial_{x_j} - (A\cdot c)_i : i=1,\dots ,n-m \rangle$. $A$-hypergeometric systems were first defined by Gel$'$fand, Graev and Zelevinsky in~\cite{GGZ}, and their systematic analysis was started by Gel$'$fand, Kapranov and Zelevinsky (see, for instance,~\cite{GKZ}). Saito, Sturmfels and Takayama have used Gr\"{o}bner deformations in the Weyl algebra to study $A$-hypergeometric systems (see~\cite{SST}). In this article, we will extend this approach to the case of Horn systems. Gel$'$fand, Graev and Retakh have also considered the {\em hypergeometric system associated with the lattice $L_{{\mathcal{B}}} = \{ {\mathcal{B}} \cdot z : z \in \Z^m \}$}, which is defined to be the left $D_n$-ideal: \[ I_{{\mathcal{B}}} + \langle A\cdot \theta - A\cdot c \rangle \subseteq D_n.\] We now introduce the $D_n$-ideal $H_{{\mathcal{B}}}(c)$, that is very closely related to the Horn system $\Horn({{\mathcal{B}}},c)$: \[H_{{\mathcal{B}}}(c) = I + \langle A \cdot \theta - A\cdot c \rangle \subseteq D_n .\] The results in Section~\ref{sec:translation} imply that, for generic $c$, there is a vector space isomorphism between the solution spaces of $\Horn({\mathcal{B}},c)$ and~$H_{{\mathcal{B}}}(c)$. Thus, we have two points of view to study Horn hypergeometric functions. We also call~$H_{{\mathcal{B}}}(c)$ a Horn system, when the context is clear. \begin{remark} We have defined the Horn operators using falling factorials because this formulation will make clearer the relationship between $\Horn({\mathcal{B}},c)$ and $H_{{\mathcal{B}}}(c)$, but it is just as legal to define Horn systems using rising factorials, as it is done in many classical sources. For instance, the Horn and Appell systems from the previous section naturally lend themselves to a rising factorial formulation. This is not really a difficulty, since switching between rising and falling factorials in the definition of Horn systems is a matter of shifting the parameters by integers. \end{remark} It is a well known result of Adolphson~\cite{adolphson} that, for generic parameters~$A\cdot c$, the holonomic rank of the $A$-hypergeometric system equals the normalized volume~$\vol(A)$ of the convex hull of the columns of~$A$, which is also the degree of the toric ideal~$I_A$. Our goal is to obtain an explicit expression in this spirit for bivariate Horn systems. Previous work in this direction required very strong assumptions (see~\cite{timur}). \begin{definition}\label{def:index} In the case that $m=2$, we set \[ \nu_{ij} = \left\{ \begin{array}{ll} \min(|b_{i1}b_{j2}|,|b_{j1}b_{i2}|), \quad & \mbox{if $b_i$, $b_j$ are in the interior of opposite quadrants of $\Z^2$}, \\ 0 & \mbox{otherwise}, \end{array} \right.\] for $1 \leq i,j\leq n$. The number $\nu_{ij}$ is called the {\em index} associated to $b_i$ and $b_j$. \end{definition} The following is the main result in this article, which follows from Theorems~\ref{thm:hbc-regular-holo},~\ref{thm:rank-formula} and~\ref{thm:horn-regular-holo}. \begin{theorem} \label{thm:main-theorem} Let ${\mathcal{B}}$ be an $n \times 2$ integer matrix of full rank such that its rows $b_1,\dots, b_n$ satisfy $b_1+\cdots+b_n=0$. If $c\in \C^n$ is a generic parameter vector, then the ideals $\Horn({\mathcal{B}},c)$ and $H_{{\mathcal{B}}}(c)$ are regular holonomic. Moreover, \[ \rank(H_{{\mathcal{B}}}(c)) = \rank(\Horn({\mathcal{B}},c)) = d_1d_2 - \sum \nu_{ij} \; ,\] where the sum runs over all pairs of linearly dependent rows~$b_i$,~$b_j$ of~${\mathcal{B}}$ lying in opposite open quadrants of~$\Z^2$. \end{theorem} We can also give an explicit basis for the solution space of $\Horn({{\mathcal{B}}},c)$ (and of $H_{{\mathcal{B}}}(c)$) (Theorem \ref{thm:sol-basis}), and compute the exact dimension of the subspace of Puiseux polynomial solutions (Theorem \ref{thm:puiseux}). %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Some observations about Horn systems} The Horn system $\Horn({\mathcal{B}},c)$ is always compatible, even if~$c$ is not generic, in the sense that its solution space is always nonempty. First of all, the constant zero function is always a solution of $\Horn({\mathcal{B}},c)$, since this system is homogeneous. Moreover, as we will see in Section \ref{sec:translation}, all the solutions of the $A$-hypergeometric system $H_A(A\cdot c)$ are solutions of $H_{{\mathcal{B}}}(c)$, and these can be transformed into solutions of $\Horn({\mathcal{B}},c)$ (see Corollary \ref{coro:vector-space-iso}), so that, under the assumptions that ${{\mathcal{B}}}$ is $n\times m$ of full rank $m$, $n>m$, with all column sums equal to zero, $\Horn({\mathcal{B}},c)$ always has nonzero solutions, since $H_A(A\cdot c)$ always has nonzero solutions (its solution space has dimension at least $\deg(I_A)=\vol(A)$, see \cite[Theorem 3.5.1]{SST}). It is easy to understand how the Horn system $\Horn({\mathcal{B}},c)$ changes if we choose a new parameter vector~$c'$, as long as $A\cdot c' = A\cdot c$. As a matter of fact, if $c = c' + {\mathcal{B}}\cdot z$, for some $z \in \C^m$, then it is easy to see that~$f(y)$ is a solution of $\Horn({\mathcal{B}},c')$ if and only if $y^z f(y)$ is a solution of $\Horn({\mathcal{B}},c)$. Notice also that the system $H_{{\mathcal{B}}}(c)$ depends only on $A\cdot c$, so that $H_{{\mathcal{B}}}(c)=H_{{\mathcal{B}}}(c')$ if $A\cdot c = A\cdot c'$. A change in~$A\cdot c\ $ can, instead, alter the solution space of $\Horn({\mathcal{B}},c)$ (and $H_{{\mathcal{B}}}(c)$) in dramatic ways. For instance, it could become infinite-dimensional, as the following example shows. \begin{example} The Horn system defined by the operators \begin{equation} (\theta_{y_{1}} + \theta_{y_{2}} + c_{1}) \theta_{y_{i}} - y_{i} (\theta_{y_{1}} + \theta_{y_{2}} + c_{2}) (\theta_{y_{1}} + \theta_{y_{2}} + c_{3}), \,\,\,\,\, i=1,2 \label{eqn:nonholo2dim} \end{equation} is not holonomic if $(c_{1} - c_{2})(c_{1} - c_{3})=0.$ Indeed, a holonomic system of equations can only have a finite-dimensional space of analytic solutions. However, since for $(c_{1} - c_{2})(c_{1} - c_{3})=0$ the operator $\theta_{y_{1}} + \theta_{y_{2}} + c_{1}$ can be factored out of each of the operators in~(\ref{eqn:nonholo2dim}), it follows that any function which is annihilated by $\theta_{y_{1}} + \theta_{y_{2}} + c_{1}$ is a solution to~(\ref{eqn:nonholo2dim}). Thus for any smooth univariate function~$u$ the product~$y_{2}^{-c_{1}}u(y_{1}/y_{2})$ satisfies~(\ref{eqn:nonholo2dim}). Notice that for generic values of the parameters $c_{1},c_{2},c_{3}$ the system~(\ref{eqn:nonholo2dim}) is holonomic. One of its solutions is given by the Gauss function $F[c_{2},c_{3};c_{1}; y_{1} + y_{2}].$ Of course, similar examples can be given in any dimension. \end{example} We could also ask what happens if we choose another matrix~${\mathcal{B}}'$ such that $A \cdot {\mathcal{B}}' = 0$. Even if~$g=g'=1$, so that~${\mathcal{B}}$ and~${\mathcal{B}}'$ are two Gale duals of~$A$, the associated Horn systems could have different holonomic rank, as we see in Example~\ref{ex:diff-ranks}. The systematic analysis of this question, in the case when~$m=2$ is one of the main objectives of this article. \begin{example} \label{ex:diff-ranks} We choose: \[ A = \left( \begin{array}{cccc} 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \end{array} \right) \; , \; B = \left( \begin{array}{rr} 1 & 0 \\ -2 & 1 \\ 1 & -2 \\ 0 & 1 \end{array} \right) \; , \; B' = \left( \begin{array}{rr} 1 & 2 \\ -2 & -3 \\ 1 & 0 \\ 0 & 1 \end{array} \right) .\] Then, if~$c$ is a generic parameter vector, $\rank(\Horn({\mathcal{B}},c))=4$, and $\rank(\Horn({\mathcal{B}}',c))=6$, as a consequence of Theorem \ref{thm:main-theorem}. This can be verified for specific values of~$c$ using the computer algebra system {\sl Macaulay 2} \cite{M2}. However, by Theorem \ref{thm:splitting-series}, these two hypergeometric systems share all fully supported solutions. \end{example} Notice that the definition of $\Horn({{\mathcal{B}}},c)$ makes sense even if ${{\mathcal{B}}}$ is a square matrix, or if the rows of ${{\mathcal{B}}}$ don't add up to zero, or even if ${{\mathcal{B}}}$ does not have full rank. As a matter of fact, we will need to consider such Horn systems on our way to proving results about the case when ${{\mathcal{B}}}$ is $n\times m$ of full rank $m$, $m0} I_{ij} \big)\] where $\sqrt{I_{ij}} = \langle \partial_i, \partial_j \rangle$, and the multiplicity of each $I_{ij}$ is $\nu_{ij}$, in the sense that \[\dim_K (\C[\partial_1,\dots,\partial_n]/I_{ij})_{\langle \partial_1,\dots,\hat{\partial_i},\dots,\hat{\partial_j}, \dots,\partial_n \rangle}=\nu_{ij},\] where $K=\C(\partial_1,\dots,\hat{\partial_i},\dots,\hat{\partial_j}, \dots,\partial_n)$. \end{proposition} We then have \begin{corollary} For $d_1$, $d_2$ the degrees of the generators of $I$, \begin{equation} \label{coro:formula} d_1 \cdot d_2 - \sum_{b_i,b_j \; \mbox{\tiny dependent}} \nu_{ij} = g \cdot \vol(A)+ \sum_{b_i,b_j \; \mbox{\tiny independent}} \nu_{ij} \; , \end{equation} where the first summation runs over linearly dependent pairs $b_i$,~$b_j$ of rows of~${\mathcal{B}}$ that lie in opposite open quadrants of~$\Z^2$, and the second summation runs over linearly independent such pairs. \end{corollary} \begin{proof} The degree of the complete intersection $I$ is $d_1 d_2$. By Proposition \ref{propo:lattice-basis-description}, this number equals \[ g \cdot \deg(I_A) + \sum \nu_{ij} \;, \] where the sum runs over all pairs of rows of ${{\mathcal{B}}}$ in opposite open quadrants of $\Z^2$. Now the result follows from the fact that the degree of $I_A$ is exactly the normalized volume $\vol(A)$ of the polytope obtained by taking the convex hull of the columns of $A$ \cite[Theorem 4.16]{gb&cp}. \end{proof} The following is another result related to the primary decomposition of~$I$, which we will need in order to prove that Horn systems have regular singularities. \begin{proposition} \label{propo:alicia's-lemma} Let ${{\mathcal{B}}} \in \Z^{n \times 2}$ of rank $2$, with rows $b_1,\dots ,b_n$, that add up to zero, and $I_{{\mathcal{B}}}$, $I$, the lattice and lattice basis ideals associated to ${{\mathcal{B}}}$. For each $1 \leq i,j \leq n$, $\nu_{ij}$ is as in Definition \ref{def:index}. Set \[ \alpha_i = \left\{ \begin{array}{ll} \max_j \; \nu_{ij} \;& \mbox{if} \; b_{i1}>0, \\ 0 & \mbox{otherwise.} \end{array} \right. \] Then \[ \partial^{\alpha} I_{{\mathcal{B}}} \subseteq I \quad \mbox{in} \; \C[\partial_1,\dots ,\partial_n]. \] \end{proposition} \begin{proof} By Proposition \ref{propo:lattice-basis-description}, it is enough to prove that $\partial^{\alpha} \in \cap_{\nu_{ij}>0} I_{ij}$. Assume that $\nu_{ij}>0$. Then $b_i$ and $b_j$ lie in the interior of opposite quadrants, so that either $b_{i1}$ or $b_{j1}$ is positive, say $b_{i1}>0$, so that $\alpha_i \geq \nu_{ij}$. We will be done if we show that $ \partial_{i}^{\nu_{ij}} \in I_{ij} .$ To do this, let $\tilde{I}_{ij}$ be the localization of $I_{ij}$ at $\langle \partial_1,\dots,\hat{\partial_i},\dots,\hat{\partial_j}, \dots,\partial_n \rangle$ so that $\tilde{I}_{ij}$ is an artinian ideal of multiplicity $\nu_{ij}$ in $K[\partial_i,\partial_j]$, where $K=\C(\partial_1,\dots,\hat{\partial_i},\dots,\hat{\partial_j}, \dots,\partial_n)$. Notice that, since $\# \{1,\partial_i,\dots ,\partial_i^{\nu_{ij}} \} = \nu_{ij} + 1$, these monomials must be linearly dependent modulo $\tilde{I}_{ij}$, so we can find $g_0,\dots,g_{\nu_{ij}} \in K$ such that \[ g_0 + g_1 \partial_i + \cdots + g_{\nu_{ij}} \partial_i^{\nu_{ij}} \in \tilde{I}_{ij} .\] But the radical of $\tilde{I}_{ij}$ is $\langle \partial_i, \partial_j \rangle$, so that $g_0 = 0$. Let $l = \min_{1\leq k \leq \nu_{ij}} \{ g_k \neq 0 \}$. Then, clearing denominators, we can find polynomials $f_l,\dots , f_{\nu_{ij}}$ {\em not} involving the variables $\partial_i, \partial_j$, $f_l \neq 0$, such that \[ \partial_i^l(f_l +\cdots + f_{\nu_{ij}}\partial_i^{\nu_{ij}-l}) \in I_{ij} .\] Now, since $I_{ij}$ is primary to $\langle \partial_i,\partial_j \rangle$, and no power of $f_l +\cdots + f_{\nu_{ij}}\partial_i^{\nu_{ij}-l}$ belongs to $\langle \partial_i,\partial_j \rangle$, then $\partial_i^l$ must belong to $I_{ij}$. Since $l \leq \nu_{ij}$, we are done. \end{proof} It is an interesting fact that the multiplicities of some of the components of $I$ do not go down under Gr\"{o}bner deformation. Given $w \in \Z^n$, and $f = \sum f_{\alpha}x^{\alpha}$ a homogeneous polynomial in $\C[\partial_1,\dots ,\partial_n]$, let \[ \ini_w(f) = \sum_{w\cdot \alpha \; \mbox{\tiny maximal over}\; f_{\alpha}\neq 0} f_{\alpha}x^{\alpha}\] and define \[\ini_w(I) = \langle \ini_w(f): f \in I \setminus \{0\}\rangle. \] The ideal $\ini_w(I)$ is called the {\em initial ideal of $I$ with respect to the weight vector $w$}. It is a monomial ideal if~$w$ is generic (see~\cite{IVA} and~\cite[Chapter 15]{commalg} for more on initial ideals, especially how to compute them). \begin{lemma} \label{lemma:multiplicities} Let~$b_k$ and~$b_l$ be two linearly dependent rows of~${\mathcal{B}}$ lying in opposite open quadrants of~$\Z^2$. If $w$ is a generic weight vector, then the multiplicity of the ideal $\langle \partial_k , \partial_l \rangle$ as an associated prime of $\ini_w(I)$ is at least the index $\nu_{kl}$. \end{lemma} As a matter of fact, the multiplicities in the previous lemma will be preserved under taking initial ideals (see Corollary \ref{coro:preservation-of-multiplicity}). \begin{proof} We assume that the vector $b_l=(-r,-s)$ is a negative multiple of $b_k=(p,q)$, and suppose that $b_k$ belongs to the first open quadrant of $\Z^2$. Up to a change of sign, the case when $b_k$ belongs to another open quadrant requires the same analysis. For simplicity we assume that $p \leq r$ (so that $s \geq q$, since $ps=qr$). The ideal $I$ is generated by binomials $\partial_k^p m_1 -\partial_l^q m_2$ and $\partial_k^r m_3-\partial_l^s m_4$, where $m_1, \dots ,m_4$ are monomials that do not contain the variables $\partial_k$, $\partial_l$. We need only consider two cases: \begin{enumerate} \item[(i)] The leading monomials of the generators of $I$ are $\partial_k^pm_1$ and $\partial_s^sm_4$. \item[(ii)] The leading monomials of the generators of $I$ are $\partial_k^pm_1$ and $\partial_k^rm_3$. \end{enumerate} In case~(i), it is clear that any other initial monomial (obtained by taking $S$-pairs) will contain a power of $\partial_k$ greater than or equal to $p$ or a power of $\partial_l$ greater than or equal to $s$. This implies that $ \langle \partial_k, \partial_l \rangle$ is an associated prime of $\ini_w(I)$, with multiplicity at least $ps=\nu_{kl}$. Now suppose we are in case~(ii), that is, the initial monomials of our two binomials are $\partial_k^p m_1$ and $\partial_k^r m_3$. Choose an integer $t$ such that $r \leq tp$. The $S$-pair between $\partial_k^p m_1 -\partial_l^q m_2$ and $\partial_k^r m_3-\partial_l^s m_4$ is: \[ \partial_l^s m_1 m_4-\partial_k^{r-p} \partial_l^q m_2m_3.\] If the first monomial is the leading term, as in case~(i), we see that every initial monomial has a factor of $\partial_k^p$ or $\partial_l^s$, and we are done. Otherwise, we take the $S$-pair of $\partial_k^pm_1-\partial_l^qm_2$ and the previous binomial. We obtain, up to a sign, \[ m_1^2 m_4 \partial_l^s - m_2^2 m_3 \partial_k^{r-2p}\partial_l^{2q} .\] Again, if the leading term is the first monomial, we are done. Otherwise, we repeat. At step $t-1$ of this process, either we found the component we wanted (with the correct multiplicity) or $m_2^{t-1}m_3\partial_k^{r-(t-1)p}\partial_l^{(t-1)q}$ is the leading term of the $S$-pair \[ m_2^{t-1}m_3\partial_k^{r-(t-1)p}\partial_l^{(t-1)q}- m_1^{t-1}m_4\partial_l^s .\] We consider the two possible situations: $r=tp$ and $rm$. Consider the surjective map \begin{align*} x^{\mathcal{B}} : (\C^*)^n &\rightarrow (\C^*)^m , \\ x &\mapsto (\prod_{j=1}^n x_j^{b_{j1}},\dots , \prod_{j=1}^n x_j^{b_{jm}}) = (x^{b^{(1)}},\dots , x^{b^{(m)}}) . \end{align*} This map is open in the sense that it takes open sets to open sets. We use it to relate the operators~$T_i$ in~$n$ variables and the operators~$H_i$ in~$m$ variables, defined in Section~\ref{sec:formula}. \begin{lemma} \label{lemma:how-to-translate} Let $U \subseteq (\C^*)^n$ be a simply connected open set and let $V = x^{{\mathcal{B}}}(U)$. We choose~$U$ small enough so that~$V$ is also simply connected. Given a holomorphic function $\psi \in {\mathcal{O}}(V)$, call $\varphi = x^c \psi(x^{{\mathcal{B}}})$. Then \begin{enumerate} \item \label{it-1} $\big(\sum_{j=1}^n a_{kj}x_j \partial_{x_j} \big)(\varphi) = (A \cdot c)_k \varphi$, for $k=1, \dots ,n-m$. \item \label{it-2} $T_i (\varphi) = 0 $ for $i=1,\dots ,m$ if and only if $H_i(\psi) = 0$ for $i=1, \dots ,m$. \item \label{it-3} Moreover, for any $u={\mathcal{B}}\cdot z \in \Z^n \in L_{{\mathcal{B}}}$, and \[ H_u = \prod_{u_j>0} \prod_{l=0}^{u_j-1} (b_j \cdot \theta_y + c_j -l) - y^z \prod_{u_j < 0} \prod_{l=0}^{|u_j|-1}(b_j\cdot \theta_y +c_j-l), \] we have $T_u(\varphi)=0$ if and only if~$H_u(\psi)=0$. \end{enumerate} \end{lemma} \begin{proof} The verifications of the three assertions are very similar. The main ingredients are the following identities: \begin{equation} \label{eqn:use-1} \theta_{x_i} x^c = x^c ( \theta_{x_i}+c_i), \quad \mbox{(in $D_n$)} , \end{equation} \begin{equation} \label{eqn:use-2} \theta_{x_i}(\psi(x^{{\mathcal{B}}}))(x) = \big[ (b_i \cdot \theta_y) \psi \big] (x^{{\mathcal{B}}}) , \end{equation} which are easily checked. Let us prove (\ref{it-2}). Call $\tilde{T}_i = \prod_{b_{ji}>0} x_j^{b_{ji}} T_i$. We have: \begin{equation} \label{eqn:star} \tilde{T}_i = \prod_{b_{ji}>0} x_j^{b_{ji}} \prod_{b_{ji}>0} \partial_{x_j}^{b_{ji}}- (x^{{\mathcal{B}}})_i \prod_{b_{ji}<0} x_j^{-b_{ji}} \prod_{b_{ji}<0} \partial_{x_j}^{-b_{ji}} . \end{equation} Recall that $(x^{{\mathcal{B}}})_i = \prod_{j=1}^n x_j^{b_{ji}}$. Using the identity: \[ x^{\alpha} \partial_x^{\alpha} = \prod_{j=1}^n \prod_{l=0}^{\alpha_j-1} (\theta_{x_j}-l)\; ,\] equation (\ref{eqn:star}) is transformed into: \[ \tilde{T}_i = \prod_{b_{ji}>0} \prod_{l=0}^{b_{ji}-1}(\theta_{x_j}-l) - (x^{{\mathcal{B}}})_i \prod_{b_{ji}<0} \prod_{l=0}^{-b_{ji}-1} (\theta_{x_j}-l) \] Using (\ref{eqn:use-1}), \begin{align*} \tilde{T}_i(\varphi)& = \tilde{T}_i(x^c \psi(x^{{\mathcal{B}}})) \\ & = x^c\bigg( \prod_{l=0}^{b_{ji}-1}(\theta_{x_j}+c_j-l) - (x^{{\mathcal{B}}})_i \prod_{b_{ji}<0} \prod_{l=0}^{-b_{ji}-1} (\theta_{x_j}+c_j-l) \bigg) (\psi(x^{{\mathcal{B}}})). \end{align*} Now (\ref{eqn:use-2}) implies that \begin{align*} \tilde{T}_i(\varphi) & = x^c\bigg( \prod_{l=0}^{b_{ji}-1}(b_j \cdot \theta_{y}(\psi)+c_j-l) - \\ & \qquad - (x^{{\mathcal{B}}})_i \prod_{b_{ji}<0} \prod_{l=0}^{-b_{ji}-1} (b_j \cdot \theta_{y}(\psi)+c_j-l) \bigg)(x^{{\mathcal{B}}}) \\ & = x^c H_i (\psi)((x^{{\mathcal{B}}}) \; . \end{align*} This shows that $\tilde{T}_i(\varphi)$ is identically zero if and only if $H_i(\psi)(x^{{\mathcal{B}}}) = 0$ for all $x \in U$. This is equivalent to~$H_i(\psi)$ vanishing identically on~$V$. Since $T_i \varphi = 0$ if and only if $\tilde{T}_i \varphi = 0$, we obtain the desired result. \end{proof} Parts (\ref{it-1}) and (\ref{it-2}) of Lemma \ref{lemma:how-to-translate} have the following consequence. \begin{corollary} \label{coro:vector-space-iso} The map \[ \begin{array}{rcl} \{ \mbox{Holomorphic solutions of} \;\; \Horn({{\mathcal{B}}},c) \; \mbox{on} \; V \} & \longrightarrow & \{ \mbox{Holomorphic solutions of} \; H_{{\mathcal{B}}}(c) \; \mbox{on} \; U \} \\ \psi & \longmapsto & x^c\psi(x^{{\mathcal{B}}}) \end{array} \] is a vector space isomorphism, that takes Puiseux polynomials to Puiseux polynomials. \end{corollary} Finally, we can use the solutions of $H_A(A\cdot c)$ to construct solutions of $H_{{\mathcal{B}}}(c)$ (and thus of $\Horn({{\mathcal{B}}},c)$). We refer to \cite[ Section 3]{SST} for background on the canonical series solutions of the $A$-hypergeometric systems introduced by Gel$'$fand, Kapranov and Zelevinsky. In the case when $c$ is generic, these canonical series solutions are fully supported logarithm-free series. \begin{theorem} \label{thm:splitting-series} Given a generic parameter vector $c$, and $\{ \phi^k : k = 1, \dots , \vol(A) \}$ a canonical basis for the space of solutions of the $A$-hypergeometric system~$H_A(A\cdot c)$, there exist linearly independent, fully supported solutions with disjoint supports \[ \{ \psi^k_l : k = 1,\dots ,\vol(A) , l = 1, \dots ,g \} \] of $\Horn({\mathcal{B}},c)$ such that \[ \phi^{k}=x^c \sum_{l=1}^g \psi^{k}_l (x^{\mathcal{B}})\; , \; \mbox{for all} \; \; k=1,\dots ,\vol(A) \; .\] Moreover, no (non trivial) linear combination of the functions~$\psi^k_l$ is ever a Puiseux polynomial. This natural decomposition holds as well for canonical series solutions with logarithms. \end{theorem} \begin{proof} By~\cite[Section 2.5]{SST} and~\cite[Proposition 5.2]{mutsumi2}, a canonical series solution $\phi$ of the $A$-hypergeometric system~$H_A(A\cdot c)$ is of the form \begin{equation} \label{eq:phi} \phi= x^{\alpha} \sum \lambda_{u,v} x^{u} \log(x^v), \end{equation} with $A \cdot \alpha = A \cdot c$, and $v, u \in L=\ker_{\Z}(A)$. We show that $\phi$ can be decomposed as a sum of $g$ solutions $\psi_1,\dots ,\psi_g$ of $H_{{\mathcal{B}}}(c)$ such that, if $\psi_j$, $\psi_l$ are nonzero, then they have disjoint supports. Observe that, if $u,v \in L$, then: \begin{equation} \label{eqn:goodtheta} \Big((A\cdot \theta)_j - (A\cdot c)_j\Big) \Big(x^{u+\alpha} \log(x^v)\Big)=0, \quad \mbox{and} \end{equation} \begin{equation} \label{eqn:no-mixing} \partial_i \Big(x^{u+\alpha} \log(x^v)\Big) = (u+\alpha)_i \ x^{u+\alpha-e_i} \log(x^v) \, + \, v_i x^{u +\alpha-e_i}. \end{equation} Consider the lattice $L_{\mathcal{B}} \subseteq \Z^n$ generated by the columns of~${\mathcal{B}}$, and its saturation~$L=\ker_{\Z}(A)$, generated by the columns of a Gale dual~$B$ of~$A$ (that is, the columns of~$B$ form a~$\Z$-basis for the integer kernel of~$A$). Let $\{ u_l : l=1,\dots ,g \}$ be a system of representatives for $L/L_{{\mathcal{B}}}$. Define \[\psi_l = x^\alpha \sum_{u \equiv u_l \mbox{\tiny mod} \;L} \lambda_{u,v} x^u \log(x^v).\] Clearly, $\phi=\psi_1+\cdots +\psi_g$, and the summands have pairwise disjoint support. By (\ref{eqn:goodtheta}), each $\psi_l$ is a solution of the system of homogeneities $\langle A\cdot \theta - A\cdot c \rangle$. Now we need to check that each $\psi_l$ is a solution of the binomial operators $T_1,\dots, T_m$ given by the columns of ${\mathcal{B}}$. Consider $T_j = \partial^{b^{(j)}_+}-\partial^{b^{(j)}_-}$. Certainly $T_j \phi = 0$. We apply the operator $T_j$ to $\phi = \psi_1+\cdots+ \psi_g$, and observe that terms coming from $T_j$ applied to $\psi_l$ cannot cancel with terms coming from $\partial^{b^{(j)}_+}$ nor from $\partial^{b^{(j)}_-}$ applied to $\psi_{l'}$ if $l \neq l'$. This is because the exponents of the monomials appearing in $\Big(\partial^{b^{(j)}_+}\Big)(\psi_l),$ for instance, are ${b^{(j)}_+}$-translates of the exponents of the monomials from $\psi_l$ by (\ref{eqn:no-mixing}), and $ {b^{(j)}_+} - {b^{(j)}_-} \in L_{{\mathcal{B}}}.$ The lack of cancellation now follows from the fact that the supports of $\psi_l$ and $\psi_{l'}$ are not congruent modulo $L_{{\mathcal{B}}}$ by construction. Now, if we have a canonical basis $\{ \phi^k, k=1, \dots, {\rm vol}(A)\}$ for the space of solutions of $H_A (A \cdot c)$ for generic $c \in \C$, they are of the form~(\ref{eq:phi}) \[\phi^k= x^{\alpha_k} \sum_{u \in L \cap {\mathcal C}_k} \lambda_{u,v} x^{u}, \] for different exponents $\alpha_k$ with respect to a generic weight vector, and~$u$ ranging over all lattice points in a full dimensional pointed cone ${\mathcal C}_k.$ Notice that, since $c$ is generic, no pair of the exponents $\alpha_k$ can differ by an integer vector. Decompose each $\phi^k = \phi^k_1 + \cdots + \phi^k_g$ as above. Note that all $\phi^k_l$ are non zero; in fact, the convex hull of all the supports is full dimensional. Moreover, the collection $\phi^k_l, \, k=1,\dots, {\rm vol}(A), \, l = 1,\dots, g$ is linearly independent since the supports are disjoint. By Lemma~\ref{lemma:how-to-translate}, each~$\phi^k_l$ is of the form $x^c \psi^{k}_l (x^{\mathcal{B}})$, where $\psi^k_l$ is a solution of $\Horn({\mathcal{B}},c)$. Clearly, no (non trivial) linear combination of the functions~$\psi^k_l$ is ever a Puiseux polynomial; in particular, they are linearly independent. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Puiseux polynomial solutions of the Horn system} \label{sec:timur} Throughout this section we assume that $m=2.$ Denote by~$\rank_p(J)$ the dimension of the space of Puiseux polynomial solutions of a $D$-ideal~$J$. Let ${\mathcal{B}}[i,j]$ be the square submatrix of~${\mathcal{B}}$ whose rows are~$b_i$ and~$b_j$, and let~$c[i,j]$ be the vector in~$\C^2$ whose coordinates are $c_i$ and~$c_j$. The first step to compute the dimension of the space of Puiseux polynomial solutions of $\Horn({\mathcal{B}},c)$ is to reduce this computation to the case when~${\mathcal{B}}$ is a $2 \times 2$ matrix. \begin{lemma} \label{lemma:ranks-split} For a generic parameter vector~$c$, \[ \rank_p(\Horn({\mathcal{B}},c)) = \sum_{i0$. If~$b_{i2}<0$, then the change of variables $\tilde{y}_1 =y_1$, $\tilde{y_2}=1/y_2$, transforms $\Horn({\mathcal{B}}[i,j],c[i,j]))$ into a Horn system given by a $2 \times 2$ matrix whose first row lies in the first open quadrant of~$\Z^2$. Thus we may assume that $b_{i1},b_{i2}>0$, and consequently $b_{j1},b_{j2}< 0$, since~$b_i$ and~$b_j$ lie in opposite open quadrants. In this case \[ H_{{\mathcal{B}}[i,j]}(c[i,j]) = \langle \partial_{i}^{b_{i1}}- \partial_{j}^{-b_{j1}}, \partial_{i}^{b_{i2}}- \partial_{j}^{-b_{j2}} \rangle ,\] and this is an ideal in the Weyl algebra with generators $x_i$,$x_j$, $\partial_i$,$\partial_j$. Let us show that any Puiseux polynomial solution~$f$ of $H_{{\mathcal{B}}[i,j]}(c[i,j])$ with irreducible support is actually a Taylor polynomial. This will imply the statement of the lemma. Choose $(u_0,v_0) \in \supp(f)$ such that $\re u_0 = \min \{ \re u : (u,v)\in \supp(f)\backslash \N^2\}$. Then $(\partial_{i}^{b_{i1}}-\partial_{j}^{-b_{j1}})f$ contains the monomial $x_i^{u_0-b_{i1}}x_j^{v_0}$ with a nonzero coefficient unless~$u_0$ is a natural number strictly less than $b_{i1}$. In this case, $v_0 \not \in \N$. Now, since all the elements of~$\supp(f)$ differ by integer vectors, and the real part of~$u_0$ is minimal, we have that $u \in \N$ for all $(u,v) \in \supp(f)\backslash \N^2$. Now pick $(u_1,v_1)$ such that the real part of~$v_1$ is minimal, and conclude that, either~$v_1$ is a natural number strictly less than~$b_{j1}$ or $x_i^{u_1}x_j^{v_1-b_{j1}}$ appears with nonzero coefficient in $(\partial_{i}^{b_{i1}}-\partial_{j}^{-b_{j1}})f=0$. But now $v \in \N$ for all $(u,v) \in \supp(f)\backslash \N^2$. We conclude that $\supp(f) \subset \N^2$. Finally, let us show that if~$b_i$ and~$b_j$ are linearly dependent, then $\Horn({{\mathcal{B}}[i,j]},c[i,j])$ only has the identically zero solution as long as~$c$ is generic. Using the change of variables $\xi_1 = y_{1}^{1/b_{i1}},$ $\xi_2 = y_{2}^{1/b_{i2}},$ we transform the operator~$b_i \cdot \theta_y$ to the operator $\theta_{\xi_1} + \theta_{\xi_2}.$ By Lemma~\ref{resintheideal} (to be proved in Section~\ref{sec:holonomic}) there exists a nonzero polynomial in~$y_1,y_2$ which lies in the ideal $\Horn({{\mathcal{B}}[i,j]},c[i,j]).$ Thus the only holomorphic solution to the system is the zero function. \end{proof} \begin{example} \rm Let us construct the Puiseux polynomial solutions to the system of equations $\Horn({\mathcal{B}},0),$ where \[ {\mathcal{B}} = \left( \begin{array}{rr} 4 & 5 \\ -3 & -5 \\ \end{array} \right). \] The system~$H_{{\mathcal{B}}}(0)$ is defined by the operators \begin{equation} \frac{\partial^{4}}{\partial x_{1}^{4}} - \frac{\partial^{3}}{\partial x_{2}^{3}}, \qquad \frac{\partial^{5}}{\partial x_{1}^{5}} - \frac{\partial^{5}}{\partial x_{2}^{5}}. \label{constantcoeff} \end{equation} Notice that we may use the parameter $0$ without loss of generality. The solutions of $H_{{\mathcal{B}}}(c)$ are exactly the same as those of $H_{{\mathcal{B}}}(0)$, and in the case of $\Horn({\mathcal{B}},c)$, the only effect is a translation of the supports of the solutions. The supports of the polynomial solutions to~(\ref{constantcoeff}) are displayed in Figure~\ref{figure1}. Two exponents are connected if the corresponding monomials are contained in a polynomial solution with irreducible support. Notice that in order to obtain these supports, we just connected the (empty) circles inside a certain rectangle to other integer points using the moves given by the columns of ${\mathcal{B}}$. %place picture here \begin{figure}[htbp] \begin{center} \psfig{file=supp1.eps} \end{center} \caption{The supports of the $15$ polynomial solutions to~(\ref{constantcoeff})} \label{figure1} \end{figure} The polynomial solutions to~(\ref{constantcoeff}) are given by \[1,\quad x_{1}, \quad x_{1}^{2}, \quad x_1^3, \quad x_2, \quad x_1x_2, \quad x_1^2x_2, \quad x_1^3x_2, \quad x_2^2, \quad x_1x_2^2, \quad x_1^2 x_2^2, \quad x_1^3x_2^2, \quad x_1^4 + 4x_2^3, \] \[x_1^4x_2 + x_2^4,\quad 5x_1^4x_2^2 + 2x_1^5 + 2x_2^5 + 40x_1x_2^3.\] Now let us unravel our isomorphism of solution spaces to obtain the corresponding solutions of $\Horn({\mathcal{B}},0)$. As in the proof of the previous lemma, if $\psi = \sum \psi_{\alpha} y^{\alpha}$ is a Puiseux polynomial solution of $\Horn({\mathcal{B}},0)$, and $\psi_{\alpha} \neq 0$, then $\binom{u}{v}={\mathcal{B}}\cdot \alpha \in \N^2$. But then \[ \alpha = {\mathcal{B}}^{-1} \cdot \binom{u}{v} = \left( \begin{array}{rr} 1 & 1 \\ -3/5 & -4/5 \\ \end{array} \right) \cdot \binom{u}{v}. \] This implies that $\alpha_1$ is a natural number, and $\alpha_2 \in (-1/5) \N$ . Moreover ${\mathcal{B}} \cdot \alpha \geq 0$. Thus, in order to find the irreducible supports of the Puiseux polynomial solutions of $\Horn({\mathcal{B}},0)$, we need to draw the region ${\mathcal{B}} \cdot \alpha \geq 0$, plot the points $\alpha \in \N \times (-1/5) \N$, and connect those points with horizontal and vertical moves. This is done in Figure~\ref{figure2}. The solid points belong to the supports of Puiseux polynomials, and the empty circles and dotted lines correspond to fully supported solutions. Thus the polynomial solutions to~$\Horn({\mathcal{B}},0)$ are as follows: \[1,\quad y_1y_2^{-3/5},\quad y_1^2y_2^{-6/5}, \quad y_1^3y_2^{-9/5}, \quad y_1y_2^{-4/5}, \quad y_1^2y_2^{-7/5}, \quad y_1^3y_2^{-2}, \quad y_1^4y_2^{-13/5}, \quad y_1^2y_2^{-8/5}, \] \[ y_1^3y_2^{-11/5}, \quad y_1^4y_2^{-14/5}, \quad y_1^5y_2^{-17/5}, \quad y_1^4y_2^{-12/5}+4y_1^3y_2^{-12/5} , \quad y_1^5y_2^{-16/5}+y_1^4y_2^{-16/5}, \] \[5 y_1^6y_2^{-4} + 2y_1^5 y_2^{-3} + 2y_1^5 y_2^{-4} + 40y_1^4y_2^{-3}.\] \begin{figure}[htbp] \begin{center} \psfig{file=supp3.eps} \end{center} \caption{The supports of the~15 Puiseux polynomial solutions to $\Horn({\mathcal{B}},0)$ in Example~\ref{ex:puiseux_pols}} \label{figure2} \end{figure} \label{ex:puiseux_pols} \end{example} We are now ready to compute $\rank_{p}(\Horn({{\mathcal{B}}[i,j]},c[i,j]))$. \begin{lemma} \label{lemma:rank-for-submatrix} The dimension of the space of Puiseux polynomial solutions of the hypergeometric system $\Horn({{\mathcal{B}}[i,j]},c[i,j])$ equals~$\nu_{ij}$ if the vectors~$b_i$ and~$b_j$ are linearly independent and lie in opposite open quadrants of~$\Z^2$. \end{lemma} \begin{proof} Suppose that~$b_i$ and~$b_j$ are linearly independent and lie in opposite open quadrants of~$\Z^2$. As in Lemma \ref{lemma:supports}, we may assume that $b_i$ lies in the interior of the first quadrant (so that $b_j$ lies in the interior of the third). By Corollary~\ref{coro:vector-space-iso}, it is sufficient to compute the number of Puiseux polynomial solutions of $H_{{\mathcal B}[i,j]}(c[i,j]).$ Introduce vectors~$\alpha$,~$\beta$ as follows: \[ \alpha = \left\{ \begin{array}{ll} (b_{i1},b_{j1}), & \;\mbox{if} \; |b_{i1}b_{j2}| > |b_{i2}b_{j1}|, \\ (-b_{i1},-b_{j1}), & \; \mbox{if} \; |b_{i1}b_{j2}| < |b_{i2}b_{j1}|, \end{array} \right. \beta = \left\{ \begin{array}{ll} (-b_{i2},-b_{j2}), & \;\mbox{if} \; |b_{i1}b_{j2}| > |b_{i2}b_{j1}|, \\ (b_{i2},b_{j2}), & \;\mbox{if} \; |b_{i1}b_{j2}| < |b_{i2}b_{j1}|. \end{array} \right.\] Furthermore, denote by~${\mathcal{R}}$ the set of points \[ {\mathcal{R}} = \left\{ \begin{array}{ll} \{ (u,v) \in \N^2 : u < b_{i2}, \, v< -b_{j1} \} , & \; \mbox{if} \; |b_{i1}b_{j2}| > |b_{i2}b_{j1}|, \\ \{ (u,v) \in \N^2 : u < b_{i1}, \, v< -b_{j2} \} , & \; \mbox{if} \; |b_{i1}b_{j2}| < |b_{i2}b_{j1}|, \end{array} \right.\] and call it the {\em base rectangle} of $H_{{\mathcal{B}}[i,j]}(c[i,j])$. By a {\em path} connecting two points $a, \tilde{a} \in \N^2$ we mean a sequence $a_1,\dots ,a_k \in \N^2$ such that $a_1=a$, $a_k=\tilde{a}$ and the difference $a_{i+1}-a_i$ is one of the vectors $\alpha$, $-\alpha$, $\beta$, $-\beta$. We say that a path is {\em increasing} if the differences are always one of~$\alpha$,~$\beta$, and that the path is {\em decreasing} the differences are always one of~$-\alpha$,~$-\beta$. We say that a point in~$\Z^2$ is connected with infinity if it can be connected with another point in~$\N^2$ which is arbitrarily far removed from the origin. Since the equations defining $H_{{\mathcal{B}}[i,j]}(c[i,j])$ can be transformed into recurrence relations for the coefficients of a polynomial solution to this system, it follows that two points can be connected by a path if and only if the monomials whose exponents are these points appear simultaneously in a polynomial solution of $H_{{\mathcal{B}}[i,j]}(c[i,j])$ that has irreducible support. Notice that if a point in~$\N^2$ is connected with infinity, then the corresponding monomial cannot be present in any polynomial solution of~$H_{{\mathcal{B}}[i,j]}(c[i,j])$. Our next observation is that there are no nonconstant increasing paths starting at a point of the base rectangle. This can be verified by direct check of all possible relations between~$|b_{i1}b_{j2}|$, $|b_{i2}||b_{j1}|$, $b_{i1}$, $b_{i2}$, $b_{j1}$, $b_{j2}$: choosing the signs of the differences $|b_{i1}b_{j2}| - |b_{i2}||b_{j1}|$, $b_{i1}-b_{i2}$, $b_{j1}-b_{j2},$ we verify this claim in each of the eight possible situations. It follows from this that no two different points in the base rectangle can be connected by a path, and that no such point is connected with infinity. Thus, any point in~$\N^2$ is either connected with a unique point in the base rectangle, or it is connected with infinity. This shows that the number of polynomial solutions of $H_{{\mathcal{B}}[i,j]}(c[i,j])$ equals the number of lattice points in ${\mathcal{R}}$, that is $\nu_{ij}=\min(|b_{i1}b_{j2}|,|b_{i2}b_{j1}|)$. \end{proof} Combining Lemmas~\ref{lemma:ranks-split} and~\ref{lemma:rank-for-submatrix}, we obtain a formula for the dimension of the space of Puiseux polynomial solutions of $\Horn({\mathcal{B}},c)$. \begin{theorem} \label{thm:puiseux} For a generic parameter~$c$, \[ \rank_p(\Horn({\mathcal{B}},c))= \sum \nu_{ij} \; ,\] where the sum runs over pairs of rows~$b_i$ and~$b_j$ of~${\mathcal{B}}$ that are linearly independent and lie in opposite open quadrants of~$\Z^2$. \end{theorem} Notice that Corollary~\ref{coro:vector-space-iso}, Theorem~3.5.1 in~\cite{SST}, Theorem~\ref{thm:puiseux}, and Theorem~\ref{thm:splitting-series} imply the following result: \begin{theorem} \label{thm:special-horn-solutions} If $c$ is generic, then \[ \rank(H_{{\mathcal{B}}}(c))= \rank(\Horn({{\mathcal{B}}},c) )\geq g \cdot \vol(A) + \sum \nu_{ij} \; ,\] where the sum runs over linearly independent rows of ${{\mathcal{B}}}$ lying in opposite open quadrants. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Regularity and holonomic rank of hypergeometric systems arising from lattices} In this section we prove that, for a generic parameter vector $\beta = A\cdot c$, the hypergeometric system $I_{\mathcal{B}}+\langle A\cdot \theta - \beta \rangle$ is regular holonomic, without restriction on the number of columns of the matrix ${{\mathcal{B}}}$. Observe that we do need our assumption that the columns of ${\mathcal{B}}$ add up to zero. Notice that genericity of $c$ implies genericity of $\beta$. In order to achieve regular holonomicity, we first notice that the holonomic property is a consequence of the fact that the fake characteristic ideal of this hypergeometric system, that is, the ideal generated by the principal symbols of the generators of $I_{{\mathcal{B}}}$ and $\langle A\cdot \theta -\beta \rangle$, already has dimension $n$. In order to prove regularity, we need to look at the hypergeometric systems arising from the primary components of $I_{{\mathcal{B}}}$. Let $\rho$ be a partial character of $L/L_{{\mathcal{B}}}$, and let $I_{\rho}$ be as in Section \ref{sec:binomialideals}. Define $H_{\rho}(A\cdot c) =I_{\rho}+\langle A\cdot \theta - A\cdot \beta \rangle$. In particular, since $\rho_0$ is the trivial character, $H_{\rho_0}(A\cdot c)=H_A(A \cdot c)$. \begin{lemma} \label{lemma:D-mod-iso} For $\rho, \rho' \in G_{{\mathcal{B}}}$, the group of partial characters of $L/L_{{\mathcal{B}}}$, the $D$-modules $H_{\rho}(\beta)$ and $H_{\rho'}(\beta)$ are isomorphic. \end{lemma} \begin{proof} It is enough to consider the case when $\rho'=\rho_0$, so that $I_{\rho'}=I_{\rho_0} = I_A$. Given any partial character $\rho: L \rightarrow \C^*$, let $p_{\rho}$ be any point in $X_{\rho}$ all of whose coordinates are nonzero. We define the map $\tau_{\rho} : D \rightarrow D$ by setting \[ \tau_{\rho}( \sum x^\alpha \partial^\beta) \, = \, \sum p_\rho^{\alpha - \beta} x^\alpha \partial^\beta.\] It is straightforward to check that $\tau_{\rho}$ defines an endomorphism of $D$, which is clearly an isomorphism. It is also easily checked that $\tau_{\rho}(I_A)=I_{\rho}$, and $\tau_{\rho}(\langle A\cdot \theta - \beta \rangle) = \langle A\cdot \theta - \beta \rangle$, so that $\tau_{\rho}(H_A(\beta))=H_{\rho}(\beta)$ and the $D$-modules $D/H_A(\beta)$ and $D/H_{\rho}(\beta)$ are isomorphic. \end{proof} \begin{corollary} \label{coro:reg-holo-Hrho} If $\rho \in G_{{\mathcal{B}}}$, the $D$-module $D/H_{\rho}(A\cdot c)$ is regular holonomic for all $c \in \C^n$. \end{corollary} \begin{proof} Hotta has shown (see \cite{equivariant}) that $D/H_A(A\cdot c)$ is regular holonomic for all parameters $c \in \C^n$, since the condition that the sum of the rows of ${{\mathcal{B}}}$ equals zero implies that the vector $(1,1,\dots,1)\in \Z^n$ belongs to the row-span of $A$. Now apply Lemma \ref{lemma:D-mod-iso}. \end{proof} We have shown that the hypergeometric systems arising from the primary components of the lattice ideal $I_{\mathcal{B}}$ are regular holonomic for all parameters. This implies that the solutions of these systems belong to the Nilsson class \cite[Ch. 6.4]{jeb1}. We will show that the solutions of the hypergeometric system $I_{{\mathcal{B}}}+ \langle A\cdot \theta - \beta \rangle$ satisfy the same properties. Recall that $I_{{\mathcal{B}}}=\cap_{\rho \in G_{\mathcal{B}}} I_{\rho}$, where $G_{\mathcal{B}}$ is the order $g$ group of partial characters, with corresponding ideals $I_{\rho}$. For any $\mathcal{J} \subseteq G_{\mathcal{B}}$, we denote by $I_{{\mathcal{J}}}$ the intersection $\cap_{\rho \in {\mathcal{J}}} I_{\rho}$. We first need the following result. \begin{proposition} \label{thm:iso} Let $w \in \N^n \backslash \{ 0 \}$. For generic $\beta$, the map \[ {D}/(I_{\mathcal{J}}+ \langle A\cdot \theta - \beta - A \cdot w \rangle ) \xrightarrow{\quad \cdot \; \partial^w \quad} {D}/(I_{\mathcal{J}}+ \langle A\cdot \theta - \beta \rangle) \:, \] given by right multiplication by $\partial^w$, is an isomorphism of left $D$-modules. \end{proposition} \begin{proof} It is sufficient to consider the case when $w = e_i$, so that our map is right multiplication by $\partial_i$. In order to use the exact argument of the proof of \cite[Theorem 4.5.10]{SST} (the analogous result for $A$-hypergeometric systems), we need to show that there exists a nonzero parametric $b$-function (see \cite[Section 4.4]{SST}), that is, we need to prove that the following elimination ideal in the polynomial ring $\C[s_1,\dots,s_d]=\C[s]$ \[ \left( D[s] \; I_{{\mathcal{J}}}+ \langle A \cdot \theta - s \rangle + D[s] \; \langle \partial_i \rangle \right) \cap \C[s] \] is nonzero, where $D[s]$ is the parametric Weyl algebra. In order to do this, we first go through an intermediate step: \begin{align*} \left( D[s] \; I_{{\mathcal{J}}}+ \langle A \cdot \theta - s \rangle + D[s] \; \langle \partial_i \rangle \right) \cap \C[\theta, s] & \supseteq \left( D[s](I_{{\mathcal{B}}}+\langle \partial_i\rangle) + \langle A \cdot \theta - s \rangle \right) \cap \C[\theta, s] \\ & = \left( D[s] (\ini_{-e_i}(I_{\mathcal{B}}+\langle \partial_i \rangle) + \langle A \cdot \theta - s \rangle \right) \cap \C[\theta,s] \\ & = \ini_{(-e_i,e_i,0)} \left(I_{\mathcal{B}}+\langle A\cdot \theta - s \rangle \right) \cap \C[\theta, s] \\ & \supseteq \langle [\theta]_{u} : \partial^u \in \ini_{-e_i}(I_{\mathcal{B}}) \rangle + \langle \theta_i \rangle + \langle A\cdot \theta - s \rangle \\ & \supseteq \langle [\theta]_{g\ u} : \partial^u \in \ini_{-e_i}(I_A) \rangle + \langle \theta_i \rangle + \langle A\cdot \theta - s \rangle \; .\\ \end{align*} Here $[\theta]_u = \prod_{k=1}^n \prod_{l=0}^{u_k-1} (\theta_k-l)$. The first containment holds because $I_{\mathcal{B}} \subseteq I_{\mathcal{J}}$. The next equality is true since \[ I_{\mathcal{B}} + \langle \partial_i \rangle = \ini_{-e_i}(I_{{\mathcal{B}}}) + \langle \partial_i \rangle.\] The equality in the third line holds by the proof of \cite[Theorem 3.1.3]{SST}, which applies here since $I_{\mathcal{B}}$ is homogeneous with respect to the multi-grading given by the columns of $A$. The next inclusion is easy to check, given that, for a monomial $\partial^u$, $x^u\partial^u = [\theta]_u$. The last containment follows from the fact that $g \ u \in L_B$ for all $u \in {\rm ker}_\Z (A)$. Now if we prove that \[ \left( \langle [\theta]_{g \ u} : \partial^u \in \ini_{-e_i}(I_A) \rangle + \langle \theta_i \rangle + \langle A\cdot \theta - s \rangle \right) \cap \C[s] \] is nonzero, we will be done. But this is a commutative elimination, so all we need to do is show that the projection of the zero set of $\langle [\theta]_{g \ u} : \partial^u \in \ini_{-e_i}(I_A) \rangle + \langle \theta_i \rangle + \langle A\cdot \theta - s \rangle$ onto the $s$-variables is not surjective. Observe that the projection of $\langle [\theta]_{u} : \partial^u \in \ini_{-e_i}(I_A) \rangle + \langle \theta_i \rangle + \langle A\cdot \theta - s \rangle$ onto the $s$-variables is not surjective (by \cite[Corollary 4.5.9]{SST}). This projection is clearly the union of affine spaces of different dimensions. But then the projection that we want is not surjective, since it is obtained from this one by adding translates of some of the affine spaces appearing in it. This concludes the proof. \end{proof} \begin{theorem} \label{thm:sols-for-lattice} For generic $\beta$, any solution $f$ of $ I_{{\mathcal{J}}}+\langle A\cdot \theta - \beta \rangle$ can be written as a linear combination \[ f = \sum_{\rho \in {\mathcal{J}}} f_{\rho} \; ,\] where $f_{\rho}$ is a solution of $I_{\rho}+\langle A\cdot \theta - \beta \rangle$. In particular, the solutions of $I_{\mathcal{B}}+\langle A\cdot \theta - \beta \rangle$ are linear combinations of the solutions of the systems $I_{\rho}+\langle A\cdot \theta - \beta \rangle$, for $\rho \in G_{{\mathcal{B}}}$. \end{theorem} \begin{proof} We proceed by induction on the cardinality of ${\mathcal{J}}$, the base case being trivial. Assume that our conclusion is valid for subsets of $G_{\mathcal{B}}$ of cardinality $r-1 \geq 1$, pick ${\mathcal{J}} \subseteq G_{\mathcal{B}}$ of cardinality $r$ and fix $\rho \in {\mathcal{J}}$. Let $P$ be an element of $I_{{\mathcal{J}} \backslash \{ \rho \} }$ such that $P \not \in I_{\rho}$. Since all of the ideals $I_{\tau}$, $\tau \in G_{\mathcal{B}}$, are homogeneous with respect to the multi-grading given by $A$, we may assume that $P$ is homogeneous, and write \[P = \lambda_1 \partial^{u^{(1)}} + \cdots + \lambda_{p-1} \partial^{u^{(p-1)}} + \partial^{w},\] where $\lambda_1,\dots ,\lambda_{p-1} \in \C$ and $A\cdot u^{(1)}=A\cdot u^{(2)}\cdots = A\cdot u^{(p-1)}=A\cdot w$. Notice that the polynomial \[\bar{P} = \lambda_1 \partial^{u^{(1)}} + \cdots + \lambda_{p-1} \partial^{u^{(p-1)}}- [ \lambda_1 \rho(u^{(1)}-w)+ \cdots + \lambda_{p-1} \rho(u^{(p-1)}-w) ]\partial^ w \] is an element of the ideal $I_{\rho}$, since this ideal is generated by all binomials of the form $\partial^{\alpha} - \rho(\alpha-\gamma) \partial^{\gamma}$, where $A\cdot \alpha = A \cdot \gamma$. To simplify the notation, set $-\lambda$ to be the coefficient of $\partial^w$ in $\bar{P}$, that is, \[ \lambda = \lambda_1 \rho(u^{(1)}-w)+ \cdots + \lambda_{p-1} \rho(u^{(p-1)}-w).\] Now let $f$ be a solution of $I_{\mathcal{J}}+\langle A\cdot \theta - \beta \rangle$, and consider the function $\bar{P} f$. For any $Q \in I_{{\mathcal{J}}\backslash \{ \rho \}}$, we have $Q \bar{P} \in I_{{\mathcal{J}}}$. This implies that $Q \bar{P} f = 0$. Furthermore, noting that $\bar{P}$ is $A$-homogeneous of multi-degree $A \cdot w$, we conclude that $\bar{P} f$ is a solution of $I_{{\mathcal{J}}\backslash \{ \rho \}} + \langle A \cdot \theta - \beta - A \cdot w \rangle$. Since $\beta$ is generic, so is $\beta + A \cdot w$, and by the inductive hypothesis we can write $\bar{P} f = \sum_{\tau \in {\mathcal{J}} \backslash \{ \rho \} } g_{\tau}$, where each $g_{\tau}$ is a solution of $I_{\tau} + \langle A \cdot \theta - \beta - A \cdot w \rangle$. By Proposition \ref{thm:iso}, $\partial^w$ induces an isomorphism between the solution spaces of $I_{\tau}+\langle A \cdot \theta - \beta \rangle$ and $I_{\tau} + \langle A \cdot \theta - \beta - A \cdot w \rangle$, so that we can find a solution $\tilde{g}_{\tau}$ of $I_{\tau}+\langle A \cdot \theta - \beta \rangle$ such that $\partial^w\tilde{g}_{\tau}=g_{\tau}$. Now \begin{align*} \bar{P} \tilde{g}_{\tau} & = \sum_{i=1}^{p-1} \lambda_i \partial^{u{(i)}} \tilde{g}_{\tau} - \lambda \partial^w \tilde{g}_{\tau} \\ & = \left(\sum_{i=1}^{p-1} \lambda_i \tau(u^{(i)}-w) - \lambda \right) g_{\tau}. \end{align*} The last equality holds because $\tilde{g}_{\tau}$ is a solution of $I_{\tau}$, and therefore $\partial^{u^{(i)}}-\tau(u^{(i)}-w)\partial^w$ annihilates it, yielding $\partial^{u^{(i)}}\tilde{g}_{\tau} = \tau(u{(i)}-w)\partial^w \tilde{g}_{\tau} = \tau(u^{(i)}-w) g_{\tau}$. Notice that the coefficient $\sum_{i=1}^{p-1} \lambda_i \tau(u^{(i)}-w) - \lambda $ is nonzero, for otherwise we could rewrite $\bar{P}$ using the sum instead of $\lambda$, and conclude that $\bar{P} \in I_{\tau}$. But we know $P \in I_{\tau}$, so $\bar{P}-P \in I_{\tau}$, a contradiction since this is a nonzero multiple of $\partial^w$, and the ideal $I_{\tau}$ contains no monomials. (The fact that $\bar{P}-P \neq 0$ follows from $\bar{P} \in I_{\rho}$ and $P \not \in I_{\rho}$). Finally define $f_{\tau} = \left(\sum_{i=1}^{p-1} \lambda_i \tau(u^{(i)}-w) - \lambda \right)^{-1} \tilde{g}_{\tau}$, so that $f_{\tau}$ is a solution of $I_{\tau} + \langle A \cdot \theta - \beta \rangle$ and \[ \bar{P} \sum_{\tau \in {\mathcal{J}}\backslash \{ \rho \}} f_{\tau} = \sum_{\tau \in {\mathcal{J}}\backslash \{ \rho \}} g_{\tau} = \bar{P} f .\] If $h = f- \sum_{\tau \in {\mathcal{J}}\backslash \{ \rho \}} f_{\tau}$, then $h$ is a solution of $I_{{\mathcal{J}}}+ \langle A \cdot \theta - \beta \rangle$ that satisfies $\bar{P} h = 0$. Now consider $Ph$. Since $P\in I_{{\mathcal{J}} \backslash \{ \rho\} }$, $Ph$ is a solution of $I_{\rho} + \langle A \cdot \theta - \beta -A\cdot w \rangle$, and a similar argument as before yields a solution $f_{\rho}$ of $I_{\rho} + \langle A \cdot \theta - \beta \rangle$ such that $Ph=Pf_{\rho}$. Let $\tilde{h} = h- f_{\rho}$, so that $f = \sum f_{\tau} + f_{\rho} + \tilde{h}$ and $P \tilde{h} = 0$. But $\bar{P}\tilde{h} = \bar{P} h - \bar{P} f_{\rho} = 0$ since $\bar{P} \in I_{\rho}$. Now $P \tilde{h} = \bar{P} \tilde{h}=0$ implies $(P-\bar{P}) \tilde{h} = 0$, so that $\partial^w \tilde{h} = 0$, because $P-\bar{P}$ is a nonzero multiple of $\partial^w$. But then $\tilde{h}$ is a solution of $I_{{\mathcal{J}}}+ \langle A \cdot \theta - \beta\rangle$ that is mapped under $\partial^w$ to the zero element in the solution space of $I_{{\mathcal{J}}} + \langle A \cdot \theta - \beta - A\cdot w \rangle$, which, using the genericity of $\beta$ and Proposition \ref{thm:iso}, implies that $\tilde{h}=0$. Thus we have obtained an expression for $f$ as a linear combination of solutions of the systems $I_{\tau}+\langle A \cdot \theta - \beta \rangle$, $\tau \in \mathcal{J}$, and the proof of the inductive step is finished. \end{proof} Considering $\mathcal{J} = G_{\mathcal{B}},$ we deduce that all solutions of $D/(I_{{\mathcal{B}}}+\langle A\cdot \theta - \beta \rangle)$ split as a sum of solutions for each $I_\rho,$ yielding a kind of converse to Theorem \ref{thm:splitting-series}. We remark that this result is not true without the genericity assumption on $\beta$, since for certain parameters (for instance for $\beta = 0$, where the constant function $1$ is a solution), the solutions to the different ideals $H_\rho(\beta)$ are not linearly independent. Observing that $D/(I_{{\mathcal{B}}}+ \langle A\cdot \theta - \beta \rangle)$ is holonomic, we deduce: \begin{theorem} \label{thm:reg-holo-IB} Suppose that ${\mathcal{B}}$ has zero column sums. The $D$-module associated with the lattice $L_{{\mathcal{B}}}$ and a parameter $\beta \in \C^d$, $D/(I_{{\mathcal{B}}}+\langle A\cdot \theta - \beta \rangle)$, is regular holonomic for generic choices of the parameter $\beta$. \end{theorem} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Holonomicity and regularity of the Horn system $H_{{\mathcal{B}}}(c)$} \label{sec:regular-holo} In this section assume that $m=2$. Our goal is to investigate for which parameters $c$ the $D$-module $D/H_{{\mathcal{B}}}(c)$ is regular holonomic. Our major tools will be the explicit description of the primary decomposition of $I$ given in Section \ref{sec:binomialideals}, the regular holonomicity of %$D/H_A(\beta)$ (see \cite{equivariant}), the hypergeometric systems associated to lattices proved in the previous section, and the fact that regular holonomic $D$-modules form an abelian category (see \cite{borel}). First let us show that $H_{{\mathcal{B}}}(c)$ is holonomic for generic $c$. \begin{theorem} \label{thm:hbc-is-holonomic} Let $m=2$ and $c$ generic parameter vector. Then $H_{{\mathcal{B}}}(c)$ is holonomic. \end{theorem} \begin{proof} Write $I = \langle \partial^{u_+}-\partial^{u_-}, \partial^{v_+}-\partial^{v_-} \rangle$, where $u$ and $v$ are the columns of ${\mathcal{B}}$. Consider first the case when ${\mathcal{B}}$ has no linearly dependent rows in opposite open quadrants of $\Z^2$. Then the ring \[\frac{\C[x_1,\dots,x_n,z_1,\dots,z_n]}{\langle z^{u_+}-z^{u_-}, z^{v_+}-z^{v_-}\rangle + \langle \sum_{j=1}^n a_{ij}x_jz_j : i=1,\dots, n-m \rangle}\] has dimension $n$ (see Lemma \ref{lemma:hsop}). Since the polynomial ring modulo the characteristic ideal of $H_{{\mathcal{B}}}(c)$ is a subring of this one, we conclude that $H_{{\mathcal{B}}}(c)$ is holonomic {\em for all} $c \in \C^m$. Now assume that ${\mathcal{B}}$ has linearly dependent rows $b_i, b_j$ in opposite open quadrants of $\Z^2$. In this case, the ideal $\langle z^{u_+}-z^{u_-}, z^{v_+}-z^{v_-}\rangle + \langle \sum a_{ij}x_jz_j : j=1,\dots ,n-m \rangle$ will have a lower-dimensional component corresponding to the vanishing of $z_i$ and $z_j$, by the results in Section \ref{sec:binomialideals} about primary decomposition of codimension $2$ lattice basis ideals. To ensure holonomicity of $H_{{\mathcal{B}}}(c)$, we will construct, for each pair $b_i$, $b_j$ of linearly dependent rows of ${\mathcal{B}}$ in opposite open quadrants of $\Z^2$, an element of the ideal $H_{{\mathcal{B}}}(c)$ that contains no $x_i$, $x_j$, $\partial_i$, $\partial_j$, and that, for generic $c$, is nonzero. The principal symbol of this element will therefore not depend on $z_i$ or $z_j$. To simplify the notation, assume $b_1$ and $b_2$ are linearly dependent in opposite open quadrants of $\Z^2$. Then the complementary square submatrix of $A$ has determinant zero, so that, by performing row and column operations, we can find $p,q \in \Q$, $r \in \C$, such that $p \,\theta_1+q\, \theta_2-r$ lies in $H_{{\mathcal{B}}}(c)$. The numbers $p$ and $q$ are rational combinations of some of the elements $a_{ij}$ of the matrix $A$, the number $r$ is a linear combination of the coordinates of the vector $c$. Also, since $b_1$ and $b_2$ are linearly dependent, we can find a nonzero element $w \in L_{\mathcal{B}}$ such that $w_1=w_2=0$. Then we can find two monomials $m_1$, $m_2$ in $\C[\partial]$ with disjoint supports, that are not divisible by either $\partial_1$ or $\partial_2$ such that $\partial_1^km_1 (\partial^{w_+}-\partial^{w_-}) \in I$ for some $k>0$ and $\partial_2^lm_2 (\partial^{w_+}-\partial^{w_-}) \in I$ for some $l>0$. This follows from the arguments that proved Proposition \ref{propo:alicia's-lemma}. Call $\mu =m_1 (\partial^{w_+}-\partial^{w_-})$ and $\lambda = m_2 (\partial^{w_+}-\partial^{w_-})$. Notice that $\mu$, $\lambda$ do not depend on $\partial_1$, $\partial_2$. Then, using $x_1^k\partial_1^k = \theta_1(\theta_1-1)\cdots (\theta_1-k+1)=[\theta_1]_k$ we see that $[\theta_1]_k \mu \in H_{{\mathcal{B}}}(c)$. Similarly, $[\theta_2]_l \lambda \in H_{{\mathcal{B}}}(c)$. Consider the left ideal in the Weyl algebra generated by: \[ p\, \theta_1+q \,\theta_2-r, [\theta_1]_k \mu, [\theta_2]_l \lambda .\] This ideal is contained in $H_{{\mathcal{B}}}(c)$. Now notice that $\theta_1$, $\theta_2$, $\lambda$ and $\mu$ are pairwise commuting elements of $D_n$. This means that we can think of $ \langle p\, \theta_1+q\, \theta_2-r, [\theta_1]_k \mu, [\theta_2]_l \lambda \rangle$ as an ideal in $\C[\theta_1,\theta_2,\partial_3,\dots ,\partial_n]$, which is a commutative subring of $D_n$. We will go one step further and think of $r$ also as an indeterminate, which commutes with $\theta_1$, $\theta_2$, $\partial_3,\dots ,\partial_n$. Finding the element of $H_{{\mathcal{B}}}(c)$ that we want has now been reduced to eliminating $\theta_1$ and $\theta_2$ from \begin{equation} \label{eqn:elimination} \langle p \, \theta_1+q\, \theta_2-r, [\theta_1]_k \mu, [\theta_2]_l \lambda \rangle \subset \C[\theta_1,\theta_2,\partial_3,\dots,\partial_n,r] . \end{equation} Since the geometric counterpart of elimination is projection, in order to check that the elimination ideal \[ \langle p\, \theta_1+q \, \theta_2-r, [\theta_1]_k \mu, [\theta_2]_l \lambda \rangle \cap \C[\lambda,\mu,r] \] is nonzero, we need to show that there exist complex numbers $\partial_3,\dots,\partial_n$ and $r$ such that, for all values of $\theta_1, \theta_2 \in \C$, the tuple $(\theta_1,\theta_2,\partial_3,\dots ,\partial_n,r)$ is not a solution of (\ref{eqn:elimination}). If $(\partial_3,\dots,\partial_n)$ is generic, the polynomials $\mu$ and $\lambda$ evaluated at that point will be nonzero. Thus, in order for $[\theta_1]_k\mu$ to vanish, $\theta_1$ must be an integer between $0$ and $k$. Analogously, $\theta_2$ must be an integer between $0$ and $l$. But then, for most values of $r$, $p\, \theta_1+q\, \theta_2-r$ is nonzero. Thus, the projection of the zero set of (\ref{eqn:elimination}) onto the $\partial_3,\dots, \partial_n, r$ coordinates is not surjective. This implies that (\ref{eqn:elimination}) contains an element $P$ that does not depend on $\theta_1$ or $\theta_2$. Notice that $P$ does depend (polynomially) on $r$, which is itself a linear combination of the coordinates of $c$. Thus, for generic $c$, $P$ will be nonzero. Now $P$ is also an element of the ideal $H_{{\mathcal{B}}}(c)$, that does not depend on $x_1$, $x_2$, $\partial_1$, $\partial_2$, and is nonzero for generic $c$. \end{proof} \begin{example} Consider the matrix \[ {\mathcal{B}} = \left( \begin{array}{rr} 1 & 2 \\ -2 & -4 \\ 1 & 1 \\ 0 & 1 \end{array} \right) .\] To prove that $H_{\mathcal{B}}(c)$ is holonomic for generic $c$, we need to find an element of $H_{\mathcal{B}}(c)$ whose principal symbol does not vanish if we set $z_1=z_2=0$. To find this element, we follow the procedure outlined in the proof of the previous theorem. The first thing we need is an element of $L_{\mathcal{B}}$ with its first two coordinates equal to zero. The vector $(0,0,-1,1)$ works. It is easy to check that $\partial_1^2\partial_3^2(\partial_3-\partial_4)$ and $\partial_2^4(\partial_3-\partial_4)$ are both elements of the lattice basis ideal $I$. We can also assume that $(2,1,0,0)$ is a row of the matrix $A$. Now what remains is to eliminate $\theta_1$ and $\theta_2$ from: \[ \langle \theta_1(\theta_1-1) \partial_3^2(\partial_3-\partial_4), \theta_2(\theta_2-1)(\theta_2-2)(\theta_2-3)(\partial_3-\partial_4), 2\theta_1+\theta_2-r \rangle \subseteq \C[\theta_1,\theta_2,\partial_3,\partial_4,r], \] where $r=2c_1+c_2$. We perform the elimination on a computer algebra system to obtain the element: \[ (2c_1+c_2)(2c_1+c_2-1)(2c_1+c_2-2)(2c_1+c_2-3)(2c_1+c_2-4)(2c_1+c_2-5) \partial_3^2(\partial_3-\partial_4) \in H_{\mathcal{B}}(c) \] whose principal symbol \[ (2c_1+c_2)(2c_1+c_2-1)(2c_1+c_2-2)(2c_1+c_2-3)(2c_1+c_2-4)(2c_1+c_2-5) z_3^2(z_3-z_4) \] does not vanish along $z_1=z_2=0$ for generic $c$. \end{example} Now we need to tackle the question of regularity. Our strategy is to reduce the problem of regularity of $H_{{\mathcal{B}}}(c)$ to proving regularity of related systems. The first step is the following result. \begin{lemma} \label{lemma:exact-sequence} Let $\alpha$ be as in Proposition \ref{propo:alicia's-lemma}. For generic $c$, the sequence \begin{equation} \label{eqn:exact-sequence} 0 \rightarrow {D}/{(I_{\mathcal{B}}+\langle A \cdot \theta - A\cdot (c+\alpha) \rangle)} \xrightarrow{\quad\cdot \;\partial^{\alpha}\quad} {D}/{H_{{\mathcal{B}}}(c)} \xrightarrow{\quad\pi\quad} {D}/{(I+\langle \partial^{\alpha}\rangle+ \langle A \cdot \theta - A\cdot c \rangle)} \rightarrow 0, \end{equation} where $\pi$ is the natural projection, is exact. \end{lemma} \begin{proof} The only part of exactness that is not clear is that right multiplication by $\partial^{\alpha}$ is injective (it is well defined since $\partial^{\alpha} I_{{\mathcal{B}}} \subseteq I$). To see this, consider the following commutative diagram: \diagram[PostScript=dvips] 0 & \rTo & {D}/ (I_{{\mathcal{B}}}+\langle A\cdot \theta - A \cdot (c+\alpha) \rangle) & \rTo^{\cdot \, \partial^{\alpha}} & {D} /(I_{{\mathcal{B}}}+\langle A\cdot \theta - A \cdot c \rangle) & \rTo 0 \\ & & & \rdTo_{\cdot \, \partial^{\alpha}} & \dTo & \\ & & & & {D} /{H_{\mathcal{B}}(c)} & \enddiagram % %\[ %\begin{array}{lcccccr} %0 & \xrightarrow{\quad \quad} & {D}/ {I_{{\mathcal{B}}}+\langle % A\cdot \theta - A \cdot (c+\alpha) \rangle} & \xrightarrow{\quad % \cdot \; \partial^{\alpha} \quad} & % {D} /{I_{{\mathcal{B}}}+\langle % A\cdot \theta - A \cdot c \rangle} & \xrightarrow{\quad \quad} & 0 % \\ %& & & & & & \\ %& & \searrow ^{ \quad \cdot \; \partial^{\alpha}} %& &\swarrow & & \\ %& & & {D} /{H_{\mathcal{B}}(c)} & & & %\end{array} \] where the %right diagonal vertical arrow is the natural inclusion. The upper row of the diagram is exact by Theorem \ref{thm:iso}, since $c$ is generic. But then the commutativity implies that the %vertical diagonal arrow is injective. \end{proof} Since the category of regular holonomic $D$-modules is an abelian category \cite[pp. 303,308,316]{borel}, the previous lemma gives us an indication of how to prove that $D/H_{{\mathcal{B}}}(c)$ is a regular holonomic $D$-module, namely, we have to show that both $D/(I_{{\mathcal{B}}} + \langle A \cdot \theta - A \cdot (c+\alpha) \rangle)$ and $D/(I+\langle \partial^{\alpha} \rangle + \langle A \cdot \theta - A \cdot (c+\alpha) \rangle)$ are regular holonomic. For generic $c$, we already know that this is true for the first module (Theorem \ref{thm:reg-holo-IB}). We will now prove that the second module is regular holonomic (Corollary \ref{coro:mono-reg-holo}). We start with an auxiliary result. \begin{lemma} \label{lemma:laura's-lemma} Let $u,v \in \N^n$ such that $\langle \partial^u,\partial^v \rangle$ is a complete intersection. If $c$ is generic, then \[ \langle \partial^u,\partial^v \rangle + \langle A \cdot \theta - A \cdot c \rangle \] is a regular holonomic system of differential equations, whose solution space has a basis of Puiseux monomials. \end{lemma} \begin{proof} It is enough to show that the system \[ \langle x^u\partial^u,x^v\partial^v \rangle + \langle A \cdot \theta - A \cdot c \rangle \] satisfies the desired properties since $x^u$ and $ x^v$ are units in $\C(x)$. Now \[ \langle x^u\partial^u,x^v\partial^v \rangle + \langle A \cdot \theta - A \cdot c \rangle = \langle [\theta]_u, [\theta]_v \rangle + \langle A \cdot \theta - A \cdot c \rangle = D \cdot F,\] where \[ [\theta]_u = \prod_{k=1}^n \prod_{l=0}^{u_k-1} (\theta_k-l) \; , \] and \[ F = \langle [\theta]_u, [\theta]_v \rangle + \langle A \cdot \theta - A \cdot c \rangle \subseteq \C[\theta] .\] This means that $D\cdot F$ is a Frobenius ideal (see \cite[Section 2.3]{SST}). By \cite[Proposition 2.3.6, Theorem 2.3.11]{SST}, if we can show that $F$ is artinian and radical, it will follow that $D \cdot F$ is holonomic, with solution space spanned by $\{ x^p : p \in {\mathcal{V}}(F)\}$, where ${\mathcal{V}}(F)$ is the zero set of the ideal $F \subseteq \C[\theta]$, and we will be done. To show that $F$ is artinian and radical, we proceed as in \cite[Theorem 3.2.10]{SST}. Let $p \in {\mathcal{V}}(F)$. Then there exist $1 \leq i < j \leq n$ such that $p_i$ and $p_j$ are nonnegative integers between zero and $\max\{u_i,v_i\}$, $\max\{u_j,v_j\}$ respectively. This follows from $[\theta]_{u}(p)=[\theta]_{v}(p)=0$ and the fact that $u$ and $v$ have disjoint supports, because $\langle\partial^u, \partial^v \rangle$ is a complete intersection. Since $c$ is generic, the minor of $A$ complementary to $\{ i,j\}$ must be nonzero (otherwise the equations $\theta_i=p_i$, $\theta_j=p_j$ and $A\cdot \theta = A\cdot c$ would be incompatible). Hence its $i$-th and $j$-th coordinates determine $p$ uniquely in ${\mathcal{V}}(F)$. \end{proof} \begin{remark} If all maximal minors of $A$ are nonzero, the above lemma holds without restriction on $c$. \end{remark} \begin{theorem} Write $I=\langle \partial^{u_+}-\partial^{u_-} ,\partial^{v_+}-\partial^{v_-} \rangle$, where $u$ and $v$ are the columns of ${\mathcal{B}}$. Let $\partial^{\alpha}$ be a monomial satisfying: \begin{equation} \label{eqn:alpha} \alpha_i > 0 \Longrightarrow u_i > 0. \end{equation} Then, for generic $c$, the $D$-ideal $I+\langle \partial^{\alpha} \rangle + \langle A\cdot \theta - A \cdot c \rangle$ has only Puiseux polynomial solutions. \end{theorem} \begin{proof} We proceed by induction on $|\alpha|=\alpha_1+\cdots+\alpha_n$, the length of $\alpha$. If $|\alpha| \leq \min\{u_i:u_i>0\}$, in particular, if $|\alpha|=1$ (recall that $|u|=0$), then $\partial^{\alpha}$ divides $\partial^{u_+}$, so that all solutions of $I+\langle \partial^{\alpha} \rangle + \langle A\cdot \theta - A \cdot c \rangle$ are solutions of $\langle \partial^{\alpha},\partial^{u_-} \rangle + \langle A\cdot \theta - A \cdot c \rangle$. But the latter ideal has only Puiseux polynomial solutions by Lemma \ref{lemma:laura's-lemma}, since $c$ is generic. Assume now that our result is true for length $s$ and let $\alpha$ be of length $s+1$ satisfying (\ref{eqn:alpha}). Choose $i$ such that $\alpha_i>0$ (and so $u_i>0$), and let $\varphi$ be a solution of $I+\langle \partial^{\alpha} \rangle + \langle A\cdot \theta - A \cdot c \rangle$. The function $\partial_i \varphi$ is a solution of $I+\langle \partial^{\alpha-e_i} \rangle + \langle A\cdot \theta - A \cdot c -A\cdot e_i \rangle$. But $|\alpha-e_i|=s$ and $c+e_i$ is still generic, so the inductive hypothesis implies that $\partial_i \varphi$ is a Puiseux polynomial. Write: \[ \partial_i \varphi = \sum_{l=0}^{N_0} g_l^{(0)} x_i^l +\sum_{l=0}^{N_1} g_l^{(1)} x_i^{\mu_1+l} + \cdots +\sum_{l=0}^{N_t} g_l^{(t)} x_i^{\mu_t+l} ,\] where the $g_l^{(k)}$ are Puiseux polynomials, constant with respect to $x_i$, $t$ is a natural number, and $\mu_1,\dots,\mu_t \in \C$ are nonintegers with noninteger pairwise differences. Then \begin{equation} \label{eqn:expr-for-varphi} \varphi = \sum_{l=0}^{N_0} g_l^{(0)} \frac{x_i^{l+1}}{l+1} +\sum_{l=0}^{N_1} g_l^{(1)} \frac{x_i^{\mu_1+l+1}}{\mu_1+l+1} + \cdots +\sum_{l=0}^{N_t} g_l^{(t)} \frac{x_i^{\mu_t+l+1}}{\mu_t+l+1} + G(x_1,\dots,\hat{x_i},\dots,x_n). \end{equation} If we prove that $G$ is a Puiseux polynomial, it will follow that so is $\varphi$, and the proof will be finished. We know that $\varphi$ is a solution of $\langle A\cdot \theta - A \cdot c \rangle$. By construction, so is $\varphi - G$. Then $G$ is a solution of $\langle A\cdot \theta - A \cdot c \rangle$. Recall that $\partial_i G=0$. We also know that $\partial^{u_+} \varphi = \partial^{u_-} \varphi$. We want to compare the coefficients of the integer powers of $x_i$ in the expressions we obtain by applying $\partial^{u_+}$ and $\partial^{u_-}$ to (\ref{eqn:expr-for-varphi}). Since we are only looking at the integer powers of $x_i$, we need only look at $\sum_{l=0}^{N_0} g^{(0)}_l(x_i^{l+1}/(l+1)) +G$. \begin{equation} \label{eqn:expr1} \partial^{u_+} \left(\sum_{l=0}^{N_0} g^{(0)}_l\frac{x_i^{l+1}}{l+1} +G \right) = \sum_{l=0}^{N_0} \frac{(l+1)l(l-1)\cdots (l+2-u_i)}{l+1} (\partial^{u_+ -u_ie_i} g^{(0)}_l) x_i^{l+1-u_i} . \end{equation} Notice that there is no $G$ in the above expression, since $\partial_i G=0$ and $u_i >0$. Also, the highest power of $x_i$ appearing in (\ref{eqn:expr1}) is $x_i^{N_0+1-u_i}$. \begin{equation} \label{eqn:expr2} \partial^{u_-} \left(\sum_{l=0}^{N_0} g^{(0)}_l\frac{x_i^{l+1}}{l+1} +G \right) = \sum_{l=0}^{N_0} (\partial^{u_-}g^{(0)}_l) \frac{x_i^{l+1}}{l+1} + \partial^{u_-}G . \end{equation} We equate the coefficients of $x_i^{l+1}$ in (\ref{eqn:expr1}) and (\ref{eqn:expr2}) to obtain: \begin{equation} \label{eqn:expr3} \frac{\partial^{u_-}g^{(0)}_l}{l+1} = (l+u_i)\cdots (l+2) \partial^{u_+-u_ie_i} g^{(0)}_{l+u_i} \; , \; \mbox{for} \; l=0,\dots ,N_0-u_i. \end{equation} If $l=N_0+1-u_i,\dots,N_0$, then $\partial^{u_-}g^{(0)}_l=0$. Also, \[\partial^{u_-} G = (u_i-1)(u_i-2)\cdots 2 \cdot 1 \cdot \partial^{u_+-u_ie_i}g^{(0)}_{u_i-1}.\] Applying $\partial^{u_-}$ to (\ref{eqn:expr3}), we see that \[ \partial^{2u_-} g^{(0)}_l = (l+u_1)\cdots(l+2)(l+1) \partial^{u_+-u_ie_i}\partial^{u_-}g^{(0)}_{l+u_i} = 0 \; , \; \mbox{for}\; l = N_0+1-2u_i,\dots,N_0-u_i.\] Applying $\partial^{u_-}$ enough times, we conclude that, if $ku_i>N_0+1,$ then $\partial^{ku_-}G=0.$ %\[ \mbox{if}\; ku_i>N_0+1, \; \mbox{then}\; \partial^{ku_-}G=0. \] But now, $G$ is a solution of $\langle \partial_i,\partial^{ku_-} \rangle + \langle A \cdot \theta - A \cdot c \rangle$, and $c$ is generic. By Lemma \ref{lemma:laura's-lemma}, $G$ is a Puiseux polynomial. \end{proof} \begin{corollary} \label{coro:mono-reg-holo} For generic $c$ and $\alpha$ satisfying (\ref{eqn:alpha}), in particular, if $\alpha$ is as in Proposition \ref{propo:alicia's-lemma}, the $D$-ideal $I+\langle \partial^{\alpha} \rangle + \langle A\cdot \theta - A \cdot c \rangle$ is regular holonomic. \end{corollary} \begin{proof} Holonomicity follows since $H_{{\mathcal{B}}}(c)$ is holonomic, and regularity is a consequence of the previous theorem. \end{proof} We can finally prove the main result in this section. \begin{theorem} \label{thm:hbc-regular-holo} Assume that all column sums of ${\mathcal{B}}$ are zero. For generic $c$, the Horn system $H_{\mathcal{B}}(c)$ is regular holonomic. \end{theorem} \begin{proof} Follows from Lemma \ref{lemma:exact-sequence}, Theorem \ref{thm:reg-holo-IB} and Corollary \ref{coro:mono-reg-holo} using the fact that regular holonomic $D$-modules form an abelian category. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Indicial ideals and holonomic ranks} \label{sec:bound} Now that we know that the Horn system $H_{\mathcal{B}}(c)$ and the hypergeometric system $I_{\mathcal{B}}+\langle A\cdot \theta - A\cdot c \rangle$ are regular holonomic for generic $c$, we can use the powerful techniques introduced in the study of $A$-hypergeometric systems to compute their ranks. Our source of references and inspiration is \cite[Chapters 1,2 and 3]{SST}. Here we assume that $m=2$, although the same arguments will work for any $m$ as long as we assume that the lattice basis ideal $I$ is a complete intersection, and that the system $H_{{\mathcal{B}}}(c)$ is regular holonomic for generic $c$. Let~$J$ be a regular holonomic left $D_n$-ideal, and~$w$ a generic weight vector. Then its indicial ideal \[ \ind_w(J)= R \cdot \ini_{(-w,w)}(J) \cap \C[\theta_1,\dots,\theta_n] \] is a a zero dimensional ideal of $\C[\theta_1,\dots , \theta_n]$, whose zeros are the so-called {\em exponents} of~$J$ with respect to~$w$. Here $R$ is the ring of linear partial differential operators with rational function coefficients. The degree of~$\ind_w(J)$, that is, the number of exponents counting multiplicities, is the dimension of the space of holomorphic solutions of~$J$ \cite[Theorem 2.3.9, Theorem 2.5.1]{SST}. Although indicial ideals are extremely useful, they are hard to get a hold of in general. However, for generic parameters, we know explicitly what the indicial ideal of an $A$-hypergeometric system is (\cite[Corollary 3.1.6]{SST}), and the same ideas work for the case of Horn systems and hypergeometric systems arising from lattices. \begin{theorem} For generic parameters~$c$, we have \[ \ind_w(H_{{\mathcal{B}}}(c)) = \big( (R \cdot \ini_w(I)) \cap \C[\theta] \big)+ \langle A \cdot \theta - A \cdot c \rangle, \] and \[ \ind_w(I_{\mathcal{B}}+\langle A\cdot \theta - A \cdot c \rangle) = \big( (R \cdot \ini_w(I_{\mathcal{B}})) \cap \C[\theta] \big)+ \langle A \cdot \theta - A \cdot c \rangle, \] \end{theorem} \begin{proof} The proof of the analogous fact for $A$-hypergeometric systems follows from \cite[Theorem 3.1.3 and Proposition 3.1.5]{SST}. But \cite[Proposition 3.1.5]{SST} carries over to the cases that interest us without any modification in its proof. Moreover the proof of \cite[Theorem 3.1.3]{SST} only uses the fact that $I_A$ is homogeneous with respect to the multi-grading given by the columns of $A$, a property that both $I$ and $I_{\mathcal{B}}$ satisfy. \end{proof} Our next goal is to compute the primary decomposition of the indicial ideals of~$H_B(c)$ and $I_{\mathcal{B}}+\langle A\cdot \theta - A\cdot c \rangle$ when~$c$ is generic. The first step is to recall the definition of certain combinatorial objects that correspond to the irreducible components of a monomial ideal in a polynomial ring. \begin{definition} Let~$M$ be a monomial ideal in the polynomial ring $\C[\partial_1, \dots ,\partial_n]$. A {\em standard pair} of~$M$ is a pair $(\partial^{\eta}, \sigma)$, where~$\sigma$ is a possibly empty subset of $\{ 1, \dots ,n \}$, that satisfies \begin{enumerate} \item $\eta_i = 0$ for all $i \in \sigma$; \item for any choice of integers~$\mu_j \geq 0$, $j \in \sigma$, the monomial $\partial^{\eta} \prod_{j \in \sigma} \partial_j^{\mu_j}$ is not in~$M$; \item for all $l \not \in \sigma$, there exist integers $\mu_l \geq 0$ and $\mu_j \geq 0$, $j \in \sigma$, such that $\partial^{\eta}\partial_l^{\mu_l}\prod_{j \in \sigma} \partial_j^{\mu_j}$ lies in~$M$. \end{enumerate} \end{definition} We denote the set of standard pairs of a monomial ideal~$M$ by~$S(M)$. By~\cite[Equation (3.2)]{StuTruVo}, \[ M =\bigcap_{(\partial^{\eta},\sigma) \in S(M)} \langle \partial_i^{\eta_i+1} : i \not \in \sigma \rangle . \] The prime ideal $\langle \partial_i : i \not \in \sigma \rangle$ is associated to~$M$ if and only if there exists a standard pair of the form $(\cdot , \sigma)$ in~$S(M)$. A standard pair $(\partial^{\eta},\sigma)$ is called {\em top dimensional} if $\langle \partial_i : i \not \in \sigma \rangle$ is a minimal associated prime of~$M$, it is called {\em embedded} otherwise. It is clear from the above formula that the degree of~$M$ is equal to the cardinality of the set of top dimensional standard pairs of~$M$. Now, since the ideals $I$ and $I_{\mathcal{B}}$ are unmixed ($I$ is a complete intersection, and the associated primes of $I_{\mathcal{B}}$ are all isomorphic to $I_A$), all of the minimal primes of all the initial ideals of~$I$ have the same dimension,~$d$ (see~\cite[Corollary 1]{KalStu}), and the same holds for $I_{\mathcal{B}}$. This means that a standard pair~$(\partial^{\eta}, \sigma)$ of either $\ini_w(I)$ or $\ini_w(I_{\mathcal{B}})$ is top dimensional if and only if~$\# \sigma = d$. Let $T(\ini_w(I))$ be the set of top dimensional standard pairs $(\partial^{\eta},\sigma)$ of~$\ini_w(I)$ such that the rows of~${{\mathcal{B}}}$ indexed by $i \not \in \sigma$ are linearly independent. Notice that if $(\partial^{\eta},\sigma)$ is a top-dimensional standard pair of $\ini_w(I_{\mathcal{B}})$, then the rows of ${\mathcal{B}}$ indexed by $i \not \in \sigma$ are linearly independent (the proof of \cite[Lemma 2.3]{shared} works for lattice ideals too). Then $T(\ini_w(I_{\mathcal{B}}))$ equals the set of top-dimensional standard pairs of $\ini_w(I_{\mathcal{B}})$. Given a standard pair in either $T(\ini_w(I))$ or $T(\ini_w(I_{\mathcal{B}}))$, and an arbitrary parameter vector~$c$, there exists a unique vector $v$ such that $A \cdot v = A \cdot c$, and $v_k=\eta_k$, $v_l=\eta_l$. Suppose that $(\partial^{\eta}, \sigma)$ is a standard pair of $\ini_w(I)$ that does not belong to~$T(\ini_w(I))$. Then either $\# \sigma < m$ or $\# \sigma = n-2$ and the columns of ${\mathcal{B}}$ corresponding to the indices not in~$\sigma$ are linearly dependent. In both of these cases, for a generic choice of~$c$, the system $A \cdot v = A \cdot c$, $v_i = \eta_i$ for $i \not \in \sigma$, has no solutions. The same holds for standard pairs not in $T(\ini_w(I_{\mathcal{B}}))$. We can now describe the primary decomposition of the indicial ideals of~$H_{{\mathcal{B}}}(c)$ and $I_{\mathcal{B}}+\langle A \cdot \theta - A \cdot c\rangle$ with respect to~$w$, in analogy to~\cite[Theorem 3.2.10]{SST}. \begin{proposition} For a generic parameter~$c$, the indicial ideal of~$H_{{\mathcal{B}}}(c)$ with respect to~$w$ equals the following intersection of maximal ideals: \begin{equation} \label{eqn:fakeind} \bigcap_{(\partial^{\eta},\sigma) \in T(\ini_w(I))} \big (\langle \theta_i - \eta_i : i \not \in \sigma \rangle + \langle A \cdot \theta - A \cdot c \rangle \big) , \end{equation} and the indicial ideal of $I_{\mathcal{B}}+\langle A\cdot \theta - A\cdot c\rangle$ equals: \begin{equation} \label{eqn:fakeind2} \bigcap_{(\partial^{\eta},\sigma) \in T(\ini_w(I_{\mathcal{B}}))} \big (\langle \theta_i - \eta_i : i \not \in \sigma \rangle + \langle A \cdot \theta - A \cdot c \rangle \big) . \end{equation} \end{proposition} \begin{proof} We prove the statement for the indicial ideal of $H_{\mathcal{B}}(c)$. The other indicial ideal is computed in exactly in the same manner. By~\cite[Corollary 3.2.3]{SST}, the indicial ideal is \[ J = \langle A \cdot \theta - A \cdot c \rangle + \bigcap_{(\partial^{\eta},\sigma) \in S(\ini_w(I))} \langle \theta_i - \eta_i: i \not \in \sigma \rangle .\] It is clear that the ideal (\ref{eqn:fakeind}) is radical. If we show that it has the same zero set as~$J$, and that~$J$ has no multiple roots, we will be done. Let~$v$ be a zero of~$J$. Then $A \cdot v = A \cdot c$, and for some $(\partial^{\eta}, \sigma) \in S(\ini_w(I))$, we have that $v_i = \eta_i$ for all $i \not \in \sigma$. Since our parameter~$c$ is generic, we must have that $(\partial^{\eta},\sigma)$ belongs to~$T(\ini_w(I))$. These are exactly the roots of the ideal (\ref{eqn:fakeind}). It also follows from the genericity of $c$ that all the zeros of~$J$ are distinct, and the proof is finished. \end{proof} Notice that the degree of~$\ini_w(I)$ is $d_1 \cdot d_2$, since it coincides with the degree of the complete intersection~$I$. Then the cardinality of the set of top dimensional standard pairs is exactly $d_1 \cdot d_2$. This and the previous proposition imply the following result. Here we use very strongly that $H_{{\mathcal{B}}}(c)$ is regular holonomic, since this property allows us to conclude that the holonomic rank of $H_{{\mathcal{B}}}(c)$ equals the degree of its indicial ideal. \begin{corollary} \label{coro:first-bound} Let~$\nu$ be the sum of the multiplicities of the minimal primes of~$\ini_w(I)$ corresponding to linearly dependent sets of two rows of~${\mathcal{B}}$. For a generic parameter vector~$c$, the degree of the fake indicial ideal is exactly $d_1 \cdot d_2 - \nu$. Therefore, \[ \rank(H_{{\mathcal{B}}}(c))= \rank(\Horn({\mathcal{B}},c)) = d_1\cdot d_2-\nu = \#T(\ini_w(I)) .\] \end{corollary} Our desired formula for the generic rank of a bivariate Horn system now follows from Proposition \ref{propo:lattice-basis-description}. \begin{theorem} \label{thm:rank-formula} For generic $c$ and $m=2$, \[ \rank(H_{{\mathcal{B}}}(c))=\rank(\Horn({{\mathcal{B}}},c))= d_1 \cdot d_2 - \sum \nu_{ij} \;,\] where the sum runs over linearly dependent rows of ${{\mathcal{B}}}$ that lie in opposite open quadrants of $\Z^2$. \end{theorem} \begin{proof} By Proposition \ref{propo:lattice-basis-description}, the sum of the multiplicities of the minimal primes of $I$ corresponding to linearly dependent rows of ${{\mathcal{B}}}$ is at least the sum of the corresponding indices $\sum \nu_{ij}$. This implies that \[ \rank(H_{{\mathcal{B}}}(c)) = \rank(\Horn({{\mathcal{B}}},c) ) \leq d_1 \cdot d_2 - \sum \nu_{ij},\] where the sum runs over linearly dependent rows of ${{\mathcal{B}}}$ lying in opposite open quadrants of $\Z^2$. The reverse inequality follows from Theorem \ref{thm:special-horn-solutions}. \end{proof} We now have an explicit description of a basis of the solution space of $\Horn({{\mathcal{B}}},c)$ (and $H_{{\mathcal{B}}}(c)$). \begin{theorem} \label{thm:sol-basis} If~$c$ is generic, the fully supported series obtained by applying the isomorphism from Corollary~\ref{coro:vector-space-iso} to the fully supported series constructed in Theorem~\ref{thm:splitting-series} and the Puiseux polynomials constructed in Theorem~\ref{thm:puiseux} form a basis for the solution space of $\Horn({{\mathcal{B}}},c)$. \end{theorem} \begin{proof} Theorem \ref{thm:splitting-series} and Corollary \ref{coro:vector-space-iso} give us $g \cdot \vol(A) + \sum \nu_{ij}$ linearly independent solutions of $H_{\mathcal{B}}(c)$ (here the sum runs over linearly independent rows of ${\mathcal{B}}$). By Theorem \ref{thm:rank-formula}, these must span the solution space of $H_{\mathcal{B}}(c)$. \end{proof} Notice that we can use Theorem \ref{thm:rank-formula} to conclude that the multiplicities of the minimal primes $\langle \partial_i , \partial_j \rangle$ of $I$ corresponding to linearly dependent rows of ${{\mathcal{B}}}$ are preserved under Gr\"{o}bner deformation. \begin{corollary} \label{coro:preservation-of-multiplicity} If $b_i$ and $b_j$ are linearly dependent rows of ${{\mathcal{B}}}$ in opposite open quadrants of $\Z^2$, and $w \in \R^n$ is a generic weight vector, then the multiplicity of $\langle \partial_i , \partial_j \rangle$ as an associated prime of $\ini_w(I)$ equals the index $\nu_{ij}$. \end{corollary} The exactly same methods that proved Theorem \ref{thm:rank-formula} will compute the rank of the hypergeometric system arising from a lattice (actually, this proof is easier, since $\# T(\ini_w(I_{\mathcal{B}})) = \deg(I_{\mathcal{B}}) = g \cdot \vol(A)$ is easier to compute than $\# T(\ini_w(I))$). Notice that here we do not need to require that $m=2$, since we know that these systems are regular holonomic without restriction on the codimension of $I_{\mathcal{B}}$. \begin{theorem} For generic $c$, \[ \rank ( I_{\mathcal{B}} +\langle A \cdot \theta - A\cdot c \rangle ) =\# T(\ini_w(I_{\mathcal{B}})) = \deg(I_{\mathcal{B}}) = g \cdot \vol(A) .\] \end{theorem} We also have an explicit description of the solution space of $I_{\mathcal{B}} +\langle A \cdot \theta - A\cdot c \rangle$ for generic $c$ as a consequence of the previous theorem and Theorem \ref{thm:sols-for-lattice}. \begin{corollary} \label{coro:sols-for-lattice-refined} For generic $c$ the solution space of $I_{\mathcal{B}} +\langle A \cdot \theta - A\cdot c \rangle$ equals the direct sum of the solution spaces of the systems $I_{\rho} +\langle A \cdot \theta - A\cdot c \rangle$, for all $\rho $ in the group $G_{\mathcal{B}}$ of partial characters. \end{corollary} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Regularity and holonomicity of $\Horn({{\mathcal{B}}},c)$} \label{sec:holonomic} Throughout this section we assume that $m=2.$ Since we do not have a $D$-module isomorphism between $H_{{\mathcal{B}}}(c)$ and $\Horn({\mathcal{B}},c)$, the regular holonomicity of $H_{{\mathcal{B}}}(c)$ does not directly prove that $\Horn({\mathcal{B}},c)$ is regular holonomic. In this section we prove that the bivariate hypergeometric system $\Horn({\mathcal{B}},c)$ is regular holonomic. We first show that the system $\Horn({{\mathcal{B}}},c)$ is holonomic for generic $c$. Recall that a system of differential equations is said to be {\it holonomic} if the dimension of its characteristic variety is the same as the dimension of the variable space. We recall that we are dealing with the system of equations defined by the hypergeometric operators \begin{equation} \begin{array}{l} H_{1} = \bm{Q}_{1}(\theta) - y_{1}\bm{P}_{1}(\theta), \\ H_{2} = \bm{Q}_{2}(\theta) - y_{2}\bm{P}_{2}(\theta). \end{array} \label{horn} \end{equation} By the definition of the Horn system (see Section~\ref{sec:formula}) the bivariate polynomials~$\bm{P}_{i},\bm{Q}_{i}$ satisfy the compatibility condition \begin{equation} R_{1}(s + e_2)R_{2}(s) = R_{2}(s + e_1)R_{1}(s), \label{compatibility} \end{equation} where $R_{i}(s) = \bm{P}_{i}(s)/\bm{Q}_{i}(s + e_i)$ and $\{ e_1,e_2 \}$ is the standard basis of~$\Z^2.$ \begin{theorem} A bivariate Horn system with generic parameters is holonomic. \label{hornholonom} \end{theorem} To prove this theorem we need some intermediate results and notation. Let~$(H_{1},H_{2}) \subset D_2$ denote the ideal generated by the hypergeometric operators defining the Horn system. By~$\sigma(P)$ we denote the principal symbol of a differential operator~$P$. This is an element of the polynomial ring $\C[y_1,y_2,z_1,z_2]$. The only case when a bivariate Horn system is not holonomic is when the principal symbols of all the operators in~$(H_{1},H_{2})$ have a nontrivial greatest common divisor (for otherwise we have two independent algebraic equations and hence the dimension of the characteristic variety of the Horn system is~2). Thus to prove holonomicity of~(\ref{horn}) it suffices to construct a family of operators in~$(H_{1},H_{2})$ such that the greatest common divisor of their principal symbols is~$1.$ By the construction of the operators in the Horn system (see Section~\ref{sec:formula}) the greatest common divisor of the principal symbols of~$H_{1}$ and~$H_{2}$ is given by a product of powers of linear forms $a y_{1}z_{1} + b y_{2}z_{2},$ where~$a,b\in\Z.$ Thus to prove Theorem~\ref{hornholonom} it suffices to show that for any~$a,b\in\Z$ such that $a y_{1}z_{1} + b y_{2}z_{2}$ divides ${\rm gcd}(\sigma(H_{1}),\sigma(H_{2}))$ there exists an operator~$T_{a,b}\in (H_{1},H_{2})$ whose principal symbol is not divisible by $a y_{1}z_{1} + b y_{2}z_{2}.$ \begin{remark} \rm For generic parameters the compatibility condition~(\ref{compatibility}) is equivalent to the relations \begin{equation} [ y_{1}\bm{P}_{1}(\theta), y_{2}\bm{P}_{2}(\theta) ] = 0, \quad (E_{2}\bm{Q}_{2})(\theta) (E_{1}E_{2}\bm{Q}_{1})(\theta) = (E_{1}\bm{Q}_{1})(\theta) (E_{1}E_{2}\bm{Q}_{2})(\theta), \label{diffcomp} \end{equation} where~$[\, , ]$ denotes the commutator of two operators, $(E_{i}^{\lambda} P)(s) = P(s + \lambda e_{i})$ and~$E_{i} = E_{i}^{1}.$ Indeed, the equalities~(\ref{diffcomp}) mean that the numerators (respectively the denominators) of the rational functions in~(\ref{compatibility}) are equal. The generic parameters assumption implies that no cancellations can occur and hence this is indeed the case. \end{remark} \begin{lemma} For any $\alpha,\beta,\gamma,\delta \in \C$ and ${P}_1(\theta),{P}_2(\theta),{Q}_1(\theta),{Q}_2(\theta)$ satisfying the relations \begin{equation} [ y_{1}{P}_{1}(\theta), y_{2}{P}_{2}(\theta) ] = 0, \quad (E_{2}{Q}_{2})(\theta) (E_{1}E_{2}{Q}_{1})(\theta) = (E_{1}{Q}_{1})(\theta) (E_{1}E_{2}{Q}_{2})(\theta), \label{diffcomp2} \end{equation} it holds that: \begin{align} \label{therelation} (\alpha (E_{2}^{-1} {Q}_{1})(\theta) - \beta y_{1}{P}_{1}(\theta)) &(\gamma {Q}_{2}(\theta) - \delta y_{2}{P}_{2}(\theta)) - \\ \nonumber &(\alpha (E_{1}^{-1}{Q}_{2})(\theta) - \beta y_{2}{P}_{2}(\theta)) (\gamma {Q}_{1}(\theta) - \delta y_{1}{P}_{1}(\theta)) = \left| \begin{array}{cc} \alpha & \beta \\ \gamma & \delta \\ \end{array} \right| \Psi, \end{align} where $\Psi= y_{1}{Q}_{2}(\theta){P}_{1}(\theta) - y_{2}{Q}_{1}(\theta){P}_{2}(\theta).$ \label{psilemma} \end{lemma} The proof of Lemma~\ref{psilemma} is a direct computation which uses the compatibility conditions~(\ref{diffcomp2}) and the Weyl algebra identity $(E_{i}^{-1}{Q}_{j})(\theta) y_{i} = y_{i} {Q}_{j}(\theta).$ Let us now consider a special case to which we will later reduce the case of an arbitrary bivariate Horn system with generic parameters. Namely, let us find a holonomicity condition for the system defined by the operators: \begin{equation} \begin{array}{l} U_{1} = f(t) {Q}_{1}(\theta) - y_{1} g(t) {P}_{1}(\theta), \\ U_{2} = f(t) {Q}_{2}(\theta) - y_{2} g(t) {P}_{2}(\theta), \end{array} \label{specialhorn} \end{equation} where~$f,g$ are arbitrary non zero univariate polynomials, $t=\theta_{1} + \theta_{2}$ and~${P}_{i},{Q}_{i}$ are arbitrary bivariate polynomials such that ${\rm deg}\, f + {\rm deg}\, {Q}_{i} = {\rm deg}\, g + {\rm deg}\, {P}_{i}$ and that~${P}_{i},{Q}_{i}$ satisfy~(\ref{diffcomp2}). Note that these relations are satisfied if $g(t) P_i, g(t) Q_i$ satisfy the equivalent relations. We assume also that~$t$ is not present in~${P}_{i}(\theta), {Q}_{i}(\theta),$ i.e., that none of the principal symbols of these operators vanish along the hypersurface $y_1 z_1 + y_2 z_2 = 0.$ Our goal is to ``eliminate~$t$'' from~(\ref{specialhorn}), i.e., to construct an operator in the ideal~$(U_{1},U_{2})$ whose principal symbol is not divisible by $\sigma(t)=y_1 z_1 + y_2 z_2.$ We do it as follows. \begin{lemma} \label{psilemma2} Let $\Psi$ be as in Lemma \ref{psilemma}. Then $ R(f(t),g(t)) \Psi \in (U_{1},U_{2})$, where~$R(f(t),g(t))$ is the resultant of~$f,g.$ \label{resintheideal} \end{lemma} \begin{proof} Let us write the polynomials~$f,g$ in the form $f(t) = \sum_{i=0}^{d} f_{i} t^{i},$ $g(t) = \sum_{i=0}^{d} g_{i} t^{i}.$ Notice that~$f,g$ do not have to be of the same degree since some of~$f_{i},d_{i}$ may be zero. Using~(\ref{therelation}), and the fact that the subring of the Weyl algebra generated by $\theta_1$ and $\theta_2$ is commutative, we conclude that for any~$j=0,\ldots,d$ \begin{align*} \sum_{i=0}^{d} \left| \begin{array}{cc} f_{j} & g_{j} \\ f_{i} & g_{i} \\ \end{array} \right| \Psi t^{i} & = \sum_{i=0}^d \Theta_{1j} (f_it^i {Q}_{2}(\theta) - y_2 g_i t^i {P}_{2}(\theta)) - \sum_{i=0}^d \Theta_{2j} (f_it^i {Q}_{1}(\theta)-y_1g_it^i{P}_{1}(\theta)) \\ & = \Theta_{1j} U_2 + \Theta_{2j} U_1 \in (U_1,U_2), \end{align*} where $\Theta_{1j} = f_j (E_{2}^{-1}{Q}_{1})(\theta) - g_j y_{1}{P}_{1}(\theta)$ and $\Theta_{2j} = f_j (E_{1}^{-1}{Q}_{2})(\theta) - g_j y_{2}{P}_{2}(\theta)$. Now clearly, \[\sum_{i=0}^{d} \left| \begin{array}{cc} f_{j} & g_{j} \\ f_{i} & g_{i} \\ \end{array} \right| \Psi t^{i} = \Psi \left| \begin{array}{cc} f_j & g_j \\ f(t) & g(t) \\ \end{array} \right|, \] so that: \[ \Psi \left| \begin{array}{cc} f_j & g_j \\ f(t) & g(t) \\ \end{array} \right| \in (U_1,U_2).\] In the trivial case when the polynomials~$f$ and~$g$ are proportional we have~$R(f(t),g(t))=0$ and the conclusion of the lemma is obviously true. If~$f$ is not proportional to~$g$ then the rank of the $2 \times (m + 1)$-matrix $ \left( \begin{array}{ccc} f_{0} \ldots f_{m} \\ g_{0} \ldots g_{m} \end{array} \right) $ equals~$2$ and hence $\Psi f(t), \Psi g(t) \in (U_{1},U_{2}).$ Since $(t-1)\Psi = \Psi t,$ it follows that $\Psi h(t) \in (U_{1},U_{2}),$ for any~$h(t) \in (f(t),g(t)),$ where~$(f(t),g(t))$ denotes the ideal in the ring of (commuting) univariate polynomials generated by~$f,g.$ It is known that the resultant of two polynomials lies in the ideal generated by these polynomials and hence $R(f(t),g(t)) \Psi \in (U_{1},U_{2}).$ The proof is complete. \end{proof} \begin{corollary} Suppose that ${\rm gcd} (\sigma(U_{1}), \sigma(U_{2}))$ is a power of $x_{1}z_{1} + x_{2}z_{2}.$ Then the hypergeometric system~(\ref{specialhorn}) is holonomic if and only if $R(f(t),g(t)) \neq 0.$ \label{specialholonom} \end{corollary} \begin{proof} Suppose that~$R(f(t),g(t)) = 0$ and let~$\zeta\in\C$ be a common root of the polynomials~$f,g.$ Since for any smooth univariate function~$h$ the product~$y_{2}^{\zeta} h(y_{1}/y_{2})$ is annihilated by the operator $t-\zeta = \theta_{1} + \theta_{2} - \zeta,$ it follows that the space of analytic solutions to~(\ref{specialhorn}) has infinite dimension. It is known that a holonomic system can only have finitely many linearly independent solutions and hence~(\ref{specialhorn}) is not holonomic in this case. On the other hand, if~$R(f(t),g(t))\neq 0,$ then by Lemma~\ref{resintheideal} the operator~$\Psi$ lies in the ideal~$(U_{1},U_{2}).$ By the assumption of the corollary the principal symbols of~$U_{1},U_{2}$ and~$\Psi$ are relatively prime and hence the system~(\ref{specialhorn}) is holonomic. \end{proof} \begin{example} Consider the system quoted in the introduction, given by the two hypergeometric operators $$ H_1 \, = \, x (\theta_x + \theta_y +a)(\theta_x+b)-\theta_x(\theta_x+\theta_y+c-1) , \\ H_2 \, = \, y (\theta_x + \theta_y +a)(\theta_y+b')-\theta_y(\theta_x+\theta_y+c-1) $$ for Appell's function~$F_1$. The operator $\Psi$ in Lemma \ref{psilemma} equals in this case $$\Psi = (x \ y) \ \Psi'\ , \quad {\rm where} \quad \Psi' = (x-y) \partial_x \partial_y -b' \partial_x + b \partial_y.$$ When $a-c+1 \not=0,$ we deduce from Lemma \ref{psilemma2} that $(x\ y) \ \Psi'$ lies in the $D$-ideal $\langle H_1, H_2 \rangle.$ In particular, all holomorphic solutions $\varphi$ of the Appell system will also satisfy $\Psi'(\varphi) =0.$ We point out that some authors add this third equation to the system (cf. for instance \cite[Page 48]{SST}). In fact, having this operator, the holonomicity of the system follows immediately. \end{example} We are now in a position to complete the proof of Theorem~\ref{hornholonom}. \vskip0.1cm \begin{proof}[Proof of Theorem~\ref{hornholonom}] Suppose that the polynomial ${\rm gcd} (\sigma(H_{1}),\sigma(H_{2}))$ vanishes along the hypersurface $a y_{1}z_{1} + b y_{2}z_{2} = 0.$ We aim to construct an operator in the ideal~$(H_{1},H_{2})$ whose principal symbol is not divisible by $a y_{1}z_{1} + b y_{2}z_{2}.$ The change of variables $\xi_{1} = y_{1}^{1/a},$ $\xi_{2} = y_{2}^{1/b}$ transforms the operator $a \theta_{y_{1}} + b \theta_{y_{2}}$ into the operator $\theta_{\xi_{1}} + \theta_{\xi_{2}}$ and the system~(\ref{horn}) into the system generated by the operators \begin{equation} \begin{array}{l} \hat{{Q}}_{1}(\theta_{\xi_{1}},\theta_{\xi_{2}}) - \xi_{1}^{a} \, \hat{{P}}_{1}(\theta_{\xi_{1}},\theta_{\xi_{2}}), \\ \hat{{Q}}_{2}(\theta_{\xi_{1}},\theta_{\xi_{2}}) - \xi_{2}^{b} \, \hat{{P}}_{2}(\theta_{\xi_{1}},\theta_{\xi_{2}}), \end{array} \end{equation} where~$\hat{{P}}_{i}(u,v)={{P}}_{i}(u/a,v/b),$ $\hat{{Q}}_{i}(u,v)={Q}_{i}(u/a,v/b).$ Let us introduce operators $\lambda_{ia}^{k},\mu_{ia}^{k}$ acting on a bivariate polynomial~$P$ as follows: \begin{equation} \lambda_{ia}^{k} (P) = \prod_{j=1}^{k} (E_{i}^{-ja} P), \qquad \mu_{ia}^{k} (P) = \prod_{j=0}^{k-1} (E_{i}^{ja} P). \label{adjustop} \end{equation} (Notice that the upper index here is {\bf not} a power.) The next Weyl algebra identities follow directly from the definition of~$\lambda_{ia}^{k},\mu_{ia}^{k}$ (the arguments of all of the involved polynomials being~$\theta_{\xi_{1}}, \theta_{\xi_{2}}$): \begin{equation} \begin{array}{rcl} \lambda_{1a}^{k} \, (\hat{{Q}}_{1}) \xi_{1}^{a} &=& \xi_{1}^{a} \, \hat{{Q}}_{1} \, \lambda_{1a}^{k-1} \, (\hat{{Q}}_{1}), \\ \lambda_{2b}^{k} \, (\hat{{Q}}_{2}) \xi_{2}^{b} &=& \xi_{2}^{b} \, \hat{{Q}}_{2} \, \lambda_{2b}^{k-1} \, (\hat{{Q}}_{2}), \\ \mu_{1a}^{k} \, (\hat{{P}}_{1}) \xi_{1}^{a} \, \hat{{P}}_{1} &=& \xi_{1}^{a} \, \mu_{1a}^{k+1} \, (\hat{{P}}_{1}), \\ \mu_{2b}^{k} \, (\hat{{P}}_{2}) \xi_{2}^{b} \, \hat{{P}}_{2} &=& \xi_{2}^{b} \, \mu_{2b}^{k+1} \, (\hat{{P}}_{2}). \end{array} \label{adjustopiden} \end{equation} Using~(\ref{adjustopiden}) we arrive at the equalities \begin{align} \label{transformedhorn1} \left( \sum_{\nu=0}^{b-1} \xi_{1}^{\nu a} \lambda_{1a}^{b-1-\nu} (\hat{{Q}}_{1})(\theta_{\xi}) \mu_{1a}^{\nu}(\hat{{P}}_{1})(\theta_{\xi}) \right) (\hat{{Q}}_{1}(\theta_{\xi}) - & \xi_{1}^{a} \hat{{P}}_{1}(\theta_{\xi})) = \\ \nonumber %\phantom{----------------} %\begin{equation} %\phantom{----------------} & \hat{{Q}}_{1}(\theta_{\xi}) \lambda_{1a}^{b-1} (\hat{{Q}}_{1})(\theta_{\xi}) - \xi_{1}^{ab} \mu_{1a}^{b} (\hat{{P}}_{1})(\theta_{\xi}), \end{align} \begin{align} \label{transformedhorn2} \left( \sum_{\nu=0}^{a-1} \xi_{2}^{\nu b} \lambda_{2b}^{a-1-\nu} (\hat{{Q}}_{2})(\theta_{\xi}) \mu_{2b}^{\nu}(\hat{{P}}_{2})(\theta_{\xi}) \right) (\hat{{Q}}_{2}(\theta_{\xi}) - & \xi_{2}^{b} \hat{{P}}_{2}(\theta_{\xi})) = \\ \nonumber &\hat{{Q}}_{2}(\theta_{\xi}) \lambda_{2b}^{a-1} (\hat{{Q}}_{2})(\theta_{\xi}) - \xi_{2}^{ab} \mu_{2b}^{a} (\hat{{P}}_{2})(\theta_{\xi}). \end{align} The differential operators~(\ref{transformedhorn1}) and (\ref{transformedhorn2}) are Horn-type hypergeometric operators in the variables~$\eta_{1}= \xi_{1}^{ab}$ and~$\eta_{2}=\xi_{2}^{ab}.$ Let us write these operators in the form $$ \begin{array}{l} \tilde{U}_{1} = f(\tau) \tilde{{Q}}_{1}(\theta_{\eta}) - \eta_{1} g(\tau) \tilde{{P}}_{1}(\theta_{\eta}), \\ \tilde{U}_{2} = f(\tau) \tilde{{Q}}_{2}(\theta_{\eta}) - \eta_{2} g(\tau) \tilde{{P}}_{2}(\theta_{\eta}), \\ \end{array} $$ where~$f,g$ are univariate polynomials, $\tau=\theta_{\eta_{1}} + \theta_{\eta_{2}}$ and none of the principal symbols of the operators $\tilde{{P}}_{i}(\theta_{\eta}), \tilde{{Q}}_{i}(\theta_{\eta})$ vanish along the hypersurface $\eta_{1} z_{1} + \eta_{2} z_{2}=0.$ %(In other words, everything that has to do with~$\tau$ is collected in %the operators~$f(\tau), g(\tau).$) The existence of such polynomials~$f,g$ follows from the compatibility condition which is satisfied by~(\ref{transformedhorn1}),(\ref{transformedhorn2}). By Lemma~\ref{resintheideal} the operator $\tilde{\Psi} = \eta_{1}\tilde{{Q}}_{2}(\theta_{\eta}) \tilde{{P}}_{1}(\theta_{\eta})- \eta_{2}\tilde{{Q}}_{1}(\theta_{\eta}) \tilde{{P}}_{2}(\theta_{\eta})$ lies in the ideal~$(\tilde{U}_{1},\tilde{U}_{2})$ as long as the parameters of the original Horn system~(\ref{horn}) are generic. Notice that by construction the principal symbol of~$\tilde{\Psi}$ does not vanish along the hypersurface $\eta_{1} z_{1} + \eta_{2} z_{2}=0.$ Going back to the variables~$y_{1},y_{2},$ we conclude that there exists an operator in~$(H_{1},H_{2})$ whose principal symbol is not divisible by $a y_{1} z_{1} + b y_{2} z_{2}.$ This completes the proof of Theorem~\ref{hornholonom}. \end{proof} We are finally ready to show that $\Horn({{\mathcal{B}}},c)$ is regular holonomic for generic $c$. \begin{theorem} \label{thm:horn-regular-holo} Assume that all column sums of ${\mathcal{B}}$ are zero. For generic parameters $c$, the bivariate Horn system $\Horn({{\mathcal{B}}},c)$ is regular holonomic. \end{theorem} \begin{proof} By Theorem \ref{hornholonom}, the Horn system $\Horn({{\mathcal{B}}},c)$ is holonomic. For generic parameter vectors $c$, we know that $H_{{\mathcal{B}}}(c)$ is regular holonomic, and Corollary~\ref{coro:vector-space-iso} gives a vector space isomorphism between the solution spaces of this system and the solutions of the Horn system that does not affect the growth behavior. \end{proof} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{The Cohen-Macaulay property as a tool to compute rank, and further research directions} Since the lattice basis ideal $I$ is a complete intersection and therefore Cohen-Macaulay, it is natural to try to apply the methods that proved that the holonomic rank $H_A(A\cdot c)$ is always $\vol(A) =\deg(I_A)$ when the underlying toric ideal $I_A$ is Cohen-Macaulay. The first evidence that these methods will not work is that the generic rank of the Horn system $H_{{\mathcal{B}}}(c)$ is {\em not} $\deg(I)= d_1 \cdot d_2$, unless we make the assumption that ${{\mathcal{B}}}$ has no linearly dependent rows in opposite open quadrants of $\Z^2$. If we follow the arguments that proved \cite[Lemma 4.3.7]{SST}, which is the main ingredient needed to prove that, when $I_A$ is Cohen-Macaulay, $\rank(H_A(A\cdot c)) = \vol(A)$ for all $c$, we see that the crucial point is whether the $n-m$ polynomials \begin{equation} \label{eqn:reg-seq} \sum_{j=1}^n a_{ij} x_j z_j \in \C[x_1,\dots ,x_n,z_1,\dots ,z_n],\; i=1, \dots ,n-m, \end{equation} form a regular sequence in $\C(x_1,\dots ,x_n)[z_1, \dots ,z_n]/I$, where here we think of $I$ as an ideal in the variables $z_1, \dots,z_n$. But if ${{\mathcal{B}}}$ has linearly dependent rows in opposite open quadrants, the polynomials (\ref{eqn:reg-seq}) do not even form a homogeneous system of parameters! \begin{lemma} \label{lemma:hsop} Let $m=2$. If $\langle A \cdot xz \rangle$ is ideal generated by the polynomials (\ref{eqn:reg-seq}), then the ideal $I+\langle A\cdot xz\rangle$ is artinian in $\C(x_1,\dots ,x_n)[z_1,\dots ,z_n]/I$, if and only if ${{\mathcal{B}}}$ has no linearly dependent rows in opposite open quadrants of $\Z^2$. \end{lemma} \begin{proof} We need to investigate the intersection of the zero locus of $\langle A \cdot xz \rangle$ over $\C(x)$ with the zero locus of $I$ over $\C(x)$. Specifically, we want to show that this intersection is a finite set if and only if ${{\mathcal{B}}}$ contains no linearly dependent rows in opposite open quadrants of $\Z^2$. We can perform this intersection irreducible component by irreducible component of $I$, recalling the primary decomposition of $I$ from Proposition \ref{propo:lattice-basis-description}. The toric irreducible components of $I$ we can deal with all at the same time: we know that $\C(x)[z]/(I_{{\mathcal{B}}}+\langle A\cdot xz \rangle)$ is zero-dimensional. That just leaves the primary components of $I$ corresponding to associated primes $\langle z_i, z_j \rangle$, where $b_i$ and $b_j$ lie in the interior of open quadrants of $\Z^2$. But now it is clear that such a component will meet the zero locus of $\langle A\cdot xz \rangle$ in an infinite set if and only if $b_i$ and $b_j$ are linearly dependent. \end{proof} As a consequence of Lemma \ref{lemma:hsop} and the arguments in \cite[Section 4.3]{SST}, we have one case when the fact that $I$ is a complete intersection will imply that the rank of $H_{{\mathcal{B}}}(c)$ does not depend on $c$. \begin{theorem} If ${{\mathcal{B}}}$ has no linearly dependent rows in opposite quadrants of $\Z^2$ then \[ \rank(H_{{\mathcal{B}}}(c)) = d_1\cdot d_2 \quad \mbox{for all}\; c \in \C^n.\] \end{theorem} Notice that this result holds even when the rows of ${\mathcal{B}}$ do not add up to zero, and is thus completely independent of regularity. Remark that the case in which no pair of (linearly dependent or not) rows lie in the interior of opposite quadrants corresponds precisely to the case in which the lattice ideal $I_B$ is a complete intersection. This agrees with the characterization in \cite{fsh}. There is another situation when we can apply the arguments from \cite[Section 4.3]{SST} to prove that a certain holonomic rank does not depend on $c$. Let $J$ be the ideal in $\C[\partial_1,\dots,\partial_n]$ obtained by saturating from $I$ the components $I_{ij}$ corresponding to linearly dependent rows of ${{\mathcal{B}}}$. Then \[ \deg(J)= d_1\cdot d_2 -\sum \nu_{ij} ,\] where the sum runs over the linearly dependent rows of ${{\mathcal{B}}}$ that lie in opposite open quadrants of $\Z^2$. As before, the methods in \cite[Section 4.3]{SST} prove the following result. \begin{lemma} If $J$ is Cohen-Macaulay, \[ \rank (J+\langle A \cdot \theta - A\cdot c \rangle) = \deg(J) . \] \end{lemma} The previous lemma and our rank formula for Horn systems have the following consequence. \begin{corollary} \label{coro:too-strong-assumptions} If $J$ is Cohen-Macaulay and $c$ is generic, the solution spaces of $H_{\mathcal{B}}(c)$ and $J+\langle A\cdot \theta - A\cdot c \rangle$ coincide. \end{corollary} We believe that Corollary \ref{coro:too-strong-assumptions} holds even when $J$ is not Cohen-Macaulay. It would be desirable to obtain an independent proof of this, since in that case we would have a proof of our rank formula in the case that $J$ is Cohen-Macaulay that does not rely on regularity. \bigskip The natural question at this point is whether we can extend arguments in Section \ref{sec:bound} to give an algebraic formula for the rank of a Horn system for any $m$. However, in order to use those methods, several ingredients are missing. First, we need to assume that the lattice basis ideal $I$ is a complete intersection, since this is not necessarily true if $m>2$. Moreover, it is not true in general that given a toric ideal $I_A$, one can find a lattice basis ideal contained in $I_A$ that is a complete intersection \cite{EC}. Moreover, we used very strongly the fact that $H_{{\mathcal{B}}}(c)$ is regular holonomic, and our $m=2$ proof is not directly generalizable to the higher $m$ case. In any case, in order to obtain an explicit rank formula in the case that $m>2$, combinatorial expressions for the multiplicities of the minimal primes of any lattice basis ideal are needed. The study of all these questions is an ongoing project of the authors. \bigskip \noindent {\bf Acknowledgments:} Work on this article began during T.~Sadykov's visit to the University of Buenos Aires in September~2001 and a subsequent meeting of the three authors in Stockholm in January 2002. We are very grateful to Mikael Passare, who made those visits possible. We thank Jan-Erik Bj\"{o}rk for all his help and patience. We would also like to thank Bernd Sturmfels and Michael Singer for inspiring conversations. Part of this work was completed while the first two authors were members at MSRI, which we also thank for its support and wonderful working atmosphere. \bibliographystyle{plain} \def\cprime{$'$} \def\cprime{$'$} \begin{thebibliography}{10} \bibitem{adolphson} Alan Adolphson. \newblock Hypergeometric functions and rings generated by monomials. \newblock {\em Duke Math. J.}, 73(2):269--290, 1994. \bibitem{jeb1} J.-E.~Bj\"ork. \newblock {\em Rings of Differential Operators.} \newblock North Holland Publishing Company, 1979. \bibitem{borel} A.~Borel, P.-P. Grivel, B.~Kaup, A.~Haefliger, B.~Malgrange, and F.~Ehlers. \newblock {\em Algebraic {${D}$}-modules}, volume~2 of {\em Perspectives in Mathematics}. \newblock Academic Press Inc., Boston, MA, 1987. \bibitem{EC} E.~Cattani. \newblock {\em Personal communication}, 2003. \bibitem{IVA} David Cox, John Little, and Donal O'Shea. \newblock {\em Ideals, varieties and algorithms}. \newblock Undergraduate Texts in Mathematics. Springer-Verlag, New York, second edition, 1997. \newblock An introduction to computational algebraic geometry and commutative algebra. \bibitem{elim-codim-2} Alicia Dickenstein and Bernd Sturmfels. \newblock Elimination theory in codimension 2. \newblock {\em J. Symbolic Comput.}, 34(2):119--135, 2002. \bibitem{commalg} David Eisenbud. \newblock {\em Commutative algebra}. \newblock Springer-Verlag, New York, 1995. \newblock With a view toward algebraic geometry. \bibitem{binomialideals} David Eisenbud and Bernd Sturmfels. \newblock Binomial ideals. \newblock {\em Duke Math. J.}, 84(1):1--45, 1996. \bibitem{erdelyi} Arthur Erd\'elyi. \newblock Hypergeometric functions of two variables. \newblock {\em Acta Math.}, 83:131--164, 1950. \bibitem{euler} Leonhard Euler. \newblock {\em Introductio in Analysis Infinitorum}, volume~1. \newblock Laussane, 1748. \bibitem{fsh} K.~G. Fischer and J. Shapiro. \newblock Mixed matrices and binomial ideals. \newblock {\em J. Pure Appl. Algebra}, 113:39--54, 1996. \bibitem{gauss} Carl~Friedrich Gauss. \newblock Disquisitiones generales circa seriem infinitam. \newblock Thesis, G\"ottingen, 1812. \newblock In {\em Ges. Werke}, G\"ottingen, 1866. \bibitem{gelfand-graev-retakh} I.~M. Gel{\cprime}fand, M.~I. Graev, and V.~S. Retakh. \newblock General hypergeometric systems of equations and series of hypergeometric type. \newblock {\em Uspekhi Mat. Nauk}, 47(4(286)):3--82, 1992. \bibitem{GGZ} I.~M. Gel{\cprime}fand, M.~I. Graev, and A.~V. Zelevinsky. \newblock Holonomic systems of equations and series of hypergeometric type. \newblock {\em Dokl. Akad. Nauk SSSR}, 295(1):14--19, 1987. \bibitem{GKZ} I.~M. Gel{\cprime}fand, M.~M. Kapranov, and A.~V. Zelevinsky. \newblock Hypergeometric functions and toric varieties. \newblock {\em Funktsional. Anal. i Prilozhen.}, 23(2):12--26, 1989. \bibitem{GKZbook} I.~M. Gel{\cprime}fand, M.~M. Kapranov, and A.~V. Zelevinsky. \newblock {\em Discriminants, Resultants, and Multidimensional Determinants}. \newblock Birkh\"auser, 1994. \bibitem{M2} Daniel R. Grayson and Michael E. Stillman. \newblock {\sl Macaulay 2}, a software system for research in algebraic geometry. \newblock Available at {\tt http://www.math.uiuc.edu/Macaulay2/}. \bibitem{latticebasis} Serkan Ho{\c{s}}ten and Jay Shapiro. \newblock Primary decomposition of lattice basis ideals. \newblock {\em J. Symbolic Comput.}, 29(4-5):625--639, 2000. \newblock Symbolic computation in algebra, analysis, and geometry (Berkeley, CA, 1998). \bibitem{equivariant} Ryoshi Hotta. \newblock Equivariant {$D$}-modules. \newblock In P.~Torasso, editor, {\em Proceedings of ICPAM Spring School in Wuhan}, Travaux en Cours. Paris, 1991. \bibitem{KalStu} Michael Kalkbrener and Bernd Sturmfels. \newblock Initial complexes of prime ideals. \newblock {\em Adv. Math.}, 116(2):365--376, 1995. \bibitem{kummer} Ernst~Eduard Kummer. \newblock {\"U}ber die hypergeometrische {R}eihe {${F}(\alpha,\beta, \gamma,x)$}. \newblock {\em J. f\"ur Math.}, 15, 1836. \bibitem{shared} Laura~Felicia Matusevich. \newblock Exceptional parameters for generic {$A$}-hypergeometric systems. \newblock {\em Int. Math. Res. Not.}, 2003(22):1225--1248, 2003. \bibitem{riemann} Georg Friedrich~Bernhard Riemann. \newblock {${P}$}-{F}unktionen. \newblock In {\em Ges. Math. Werke}, pages 67--84. G\"{o}ttingen, 1857. \newblock Republished Leipzig 1892. \bibitem{timur} T.~M. Sadykov. \newblock On the {H}orn system of partial differential equations and series of hypergeometric type. \newblock {\em Math. Scand.}, 91(1):127--149, 2002. \bibitem{mutsumi2} Mutsumi Saito. \newblock {Logarithm-free {$A$}-hypergeometric series}. \newblock {\em Duke Math. J.}, 115(1):53--73, 2002. \bibitem{SST} Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama. \newblock {\em Gr\"obner {D}eformations of {H}ypergeometric {D}ifferential {E}quations}. \newblock Springer-Verlag, Berlin, 2000. \bibitem{slater} Lucy~Joan Slater. \newblock {\em Generalized Hypergeometric Functions}. \newblock Cambridge University Press, 1966. \bibitem{gb&cp} Bernd Sturmfels. \newblock {\em Gr\"obner {B}ases and {C}onvex {P}olytopes}. \newblock American Mathematical Society, Providence, RI, 1996. \bibitem{StuTruVo} Bernd Sturmfels, Ng\^o~Vi\^et Trung, and Wolfgang Vogel. \newblock Bounds on degrees of projective schemes. \newblock {\em Math. Ann.}, 302(3):417--432, 1995. \end{thebibliography} \end{document}