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### Rev. Unión Mat. Argent. vol.49 no.2 Bahía Blanca July/Dec. 2008

Hypergeometric functions and binomials

Alicia Dickenstein

Abstract. We highlight the role of primary decomposition of binomial ideals in a commutative polynomial ring, in the description of the holonomicity, the holonomic rank, and the shape of solutions of multivariate hypergeometric differential systems of partial differential equations.

Partially supported by UBACYT X064, CONICET PIP 5617 and ANPCyT PICT 20569, Argentina

En honor a Mischa Cotlar, con afecto y admiración

1. Introduction

There have been two main directions in the study of classical hypergeometric functions. The first of these is to study the properties of a particular series, going back to Euler [Eu1748] and Gauss [Gau1812]. The second one is to find a differential equation satisfied by the hypergeometric function, and to study all the solutions of that equation, going back to Kummer [Kum1836] and Riemann [Rie1857].

Recall that a (convergent) univariate series

is called hypergeometric if the ratio between two consecutive coefficients is a rational function of . If we write with polynomials, this is equivalent to the recurrence . The main basic observation is that this recurrence can be expressed by the fact that the hypergeometric differential operator annihilates . Here denotes the operator .

In the generalization to hypergeometric functions in several variables, these two points of view suggest different approaches. A natural definition of hypergeometric power series in several variables was proposed by Horn [Hor1889Hor31]. A series

in variables with complex coefficients is hypergeometric in the sense of Horn if there exist rational functions in variables such that

 (1.1)

where denote the standard basis vectors of . As before, denote by the operator , . For all monomials and all polynomials it holds that , which implies again the following relation between the behaviour of the ratios of coefficients and the existence of hypergeometric differentials operators annihilating . If we write the rational functions as

where and are relatively prime polynomials and divides , the series satisfies the following Horn hypergeometric system of differential equations:

 (1.2)

Gelfand, Graev, Kapranov, and Zelevinsky [GGZ87GKZ89GKZ90] developed a highly interesting point of view, by "dressing" the hypergeometric functions and operators with "homogeneities". This allows to understand the properties of classical hypergeometric systems via tools in algebraic geometry and combinatorics. They associated to a configuration of lattice points spanning (which we encode as the columns of a integer matrix, also called ) and a vector , a holonomic left ideal in the Weyl algebra in variables as follows.

Definition 1.1. The -hypergeometric system (or GKZ-hypergeometric system) with exponent is the left ideal in the Weyl algebra generated by the toric operators , for all such that , and the Euler operators for .

We refer the reader to [SST00] for an account of -hypergeometric systems with emphasis on computations.

A local holomorphic function is -hypergeometric of degree if it is annihilated by . -hypergeometric systems are homogeneous (and complete) versions of classical hypergeometric systems in variables [GGR92DMS05], in the following sense. Let be a matrix whose colums span the integer kernel of , and also call the Gale dual lattice configuration given by the rows of . Consider the surjective open map

Let , be simply connected open sets and denote by the coordinates in . Given a holomorphic function , call , where . Then, , for . Moreover, for any , we have that if and only if , where and denote the following differential operators in and variables respectively:

 (1.3)
 (1.4)

where Note that system (1.2) corresponds to the choices (). On the -side, this amounts to considering only codimension many binomial operators (corresponding to the columns of , that is to a -basis of , and not the whole system of toric binomials in (1.1)).

Although the isomorphism sending to is only at the level of local holomorphic solutions and not at the level of -modules, it preserves many of the pertinent features, including the dimensions of the spaces of local holomorphic solutions and the structure of their series expansions. This point of view allowed us in [DMS05] to study the holonomicity, the holonomic rank and the persistence of Puiseux polynomial solutions to Horn systems for via the translation into a multihomogeneous setting and the study of primary components of binomial ideals in a commutative polynomial ring. This work was widely generalized in the recent article [DMM06], where we propose the definition of a binomial -module (see Definition 3.1 below) which comprises the notion of -hypergeometric system. In this note we illustrate our perspective and results via examples and extended comments.

2. Examples: Holonomic rank and binomials

Before giving more precise statements and definitions, we focus on classical examples under a modern "binomial" view.

2.1. Horn and Appel hypergeometric systems in two variables.

We consider two classical bivariate Horn systems of hypergeometric partial differential equations as in (1.2):

• The system associated to Horn hypergeometric series is:  y1(2θy1 - θy2 + a′)(2θy1 - θy2 + a′ + 1) - (-θy1 + 2θy2 + a)θy1f = 0 , y2(-θy1 + 2θy2 + a)(-θy1 + 2θy2 + a + 1) - (2θy1 - θy2 + a′)θyf = 0 ,

where are generic parameters.

• The system associated to Appell series  is:  y1(θy1 + θy2 + a)(θy1 + b) - θy1(θy1 + θy2 + c - 1)f = 0 , y2(θy1 + θy2 + a)(θy2 + b′) - θy2(θy1 + θy2 + c - 1)f = 0,

where and are generic parameters.

Recall that the holonomic rank of a system of linear differential operators with polynomial (or holomorphic coefficients) is the vector space dimension of the space of local holomorphic solutions around a point which is not in the singular locus of the associated -module. Both systems are defined by two linear operators in two variables, so a first guess for the holonomic rank is in both cases. This is the right answer for Horn system, but Appell system has holonomic rank equal to . Moreover, Erdélyi noted in [Erd50] that, in a neighborhood of a given point, three linearly independent solutions of Horn system can be obtained through contour integral methods. He also found a fourth linearly independent solution: the Puiseux monomial . He remarked that the existence of this elementary solution was puzzling, and offered no explanation for its occurrence. Via the translation to a homogeneous system (which is part of an -hypergeometric system, as in the Introduction), we gave in [DMS05] an explanation for these facts.

Look at the binomials

in the commutative polynomial ring . The exponents are read from the (integer) coefficients of the linear forms in occuring in the system. Its zero set in has two irreducible components. One is the toric variety which is the closure of the solutions in the torus of the binomials , which has degree and corresponds to the fully supported solutions described by Erdélyi (we refer the reader to [Stu96] for background on toric varieties). The other component lies "at infinity", that is, in the union of the coordinate hyperplanes. Its multiplicity equals the intersection multiplicity at the origin of the system of binomials in variables
This multiplicity can be computed as ([DSt02])
Since the determinant of the matrix is non zero, Theorem 2.5 in [DMS05] asserts that this multiplicity equals the dimension of the space of solutions to the Horn system which have finite support. Further results in [DMS05] explain the fact that in this case one gets a (non zero) Puiseux monomial solution.

The same result explains the holonomic degree for Appell system. The translation to the homogeneous setting gives the following two binomials:

in the commutative polynomial ring . In this case we have variables since there are different integer linear forms in occuring in the two differential operators. The zero set of in has again two components. One is the toric variety which is the closure of the solutions in the torus of the binomials , which has degree and gives fully supported linearly independent solutions. The other component lies "at infinity", and it has multiplicity But this time the determinant of the matrix with rows is zero, and so for generic parameters the differential system is holonomic and this component does not contribute any new solution.

For particular choices of parameters, Appell Horn system is not holonomic. Consider the matrix

and a vector of parameters . Denote by the partial derivative with respect to the variable and let , for . We can choose the following matrix , with Gale dual :

Set . The corresponding Horn system (1.2) looks in its "binomial incarnation" as follows:

where , , , . If , then any (local holomorphic) function annihilated by the operators is a solution, for instance all monomials with satisfying . So, the space of such functions is infinite-dimensional; in fact, it has uncountable dimension. Again, this phenomenon is explained in general in [DMS05].

2.2. Mellin hypergeometric system for algebraic functions.

Given coprime integers set

and The local roots of the generic sparse polynomial

viewed as functions of the coefficients, are algebraic solutions to the associated -hypergeometric system [Ma37Bir27Stu00CDD99PT04]. In particular, any solution to this -hypergeometric system has a double homogeneity property and can therefore be considered as a function of  variables. This implies that we can arbitrarily prescribe the values of any two nonzero coefficients in without losing any essential information on the general solution to this equation. If we divide by  and then set we obtain an algebraic equation of the form

 (2.1)

A classical result of Mellin from 1921 (see [Mel21]) states that if is a local root of (2.1), then it satisfies the following system of  partial differential equations:

 (2.2)

where and .

In [DS07] we studied the solutions of the Mellin system via homogenization and translation to the binomial setting. Mellin not only observed in [Mel21] that the roots of the algebraic equation (2.1) satisfy the Mellin system (2.2), but he also made the following remark. The solution around the origin which satisfies , is given by

 (2.3)

Here and the empty product is defined to be  All other solutions around the origin have the form

where  runs through the -roots of . It is also clear that for any choice of , the function is a root of the algebraic equation
 (2.4)

where we denote . It happens that the holonomic rank of Mellin system (2.2) equals . In the effort of getting equations such that in each of them a partial derivative of with respect to each of the variables is expressed in terms of other derivatives, the Mellin system has not only the roots of the algebraic equation (2.1) as solutions, but also the roots of the associated equations (2.4).

When we add homogeneities, we are led to an isomorphism between the solutions of the Mellin system and the solutions of a Horn system in variables associated to the following matrix :

 (2.5)

Let and consider the Horn system , given by the operators defined in (1.4). The colums of generate a non saturated lattice of order and the binomials read in its columns

define a complete intersection variety with components not at infinity, given by torus translates of the toric variety associated to the matrix (which correspond to the associate algebraic equations (2.4). The degree of (and hence of all the components) is , so by [DMM06, Theorem 6.10], the holonomic rank of the Horn system equals , which explains the holonomic rank of the Mellin system. Moreover, we describe in [DS07, Theorem 4.3] the dimension of the space of algebraic solutions and the ocurrence of explicit non algebraic logarithmic solutions. This behaviour is related to the interactions among the different primary components of . The general pattern is given in [DMM06, Theorem 6.8].

3. Main definitions and results

3.1. What is a binomial -module?.

We give now the main general definition of a binomial -module, which contains the previous hypergeometric systems as special cases. The building blocks in the world of binomial -modules are the -hypergeometric systems (1.1) for different .

We consider an integer matrix  such that the cone generated by the columns of contains no lines, all of the are nonzero, and . The matrix induces a -grading of the polynomial ring by setting . An ideal of is -graded if it is generated by elements that are homogeneous for the -grading.

A binomial ideal is an ideal generated by binomials , where are column vectors and . A binomial ideal is -graded precisely when it is generated by binomials each of which satisfies either or .

The Weyl algebra of linear partial differential operators is also -graded by additionally setting . For each , the -th Euler operator is defined as

Given a vector , we write for the sequence Note that these operators are -homogeneous of degree . A (local) holomorphic function is annihilated by precisely when is -homogeneous of homogeneity , that is

where , with all close to .

Definition 3.1. Given an -graded binomial ideal , we denote by the left ideal in the Weyl algebra . The binomial -module associated to  is the quotient .

Thus, a binomial D-module is a quotient by a left -ideal generated by an -graded binomial ideal with constant coefficients plus Euler operators of order associated to the row span of , which prescribe -homogeneity infinitesimally. As we pointed out in the classical case, the binomial differential operators annihilate a (multivariate Puiseux) series if and only if the coefficients of the series satisfy (special) linear recurrences.

When equals the toric ideal , the associated binomial -module is just the -hypergeometric system (1.1). On the other side, when is generated by binomials , , with linearly independent over (in this case is called a lattice basis ideal), the corresponding binomial -module is a Horn system (in binomial version).

We summarize the main questions concerning binomial -modules:

• For which parameters does the space of local holomorphic solutions around a nonsingular point of a binomial Horn system have finite dimension as a complex vector space?
• What is a combinatorial formula for the minimum holonomic rank, over all possible choices of parameters?
• Which parameters are generic, in the sense that the minimum dimension is attained?
• How do (the supports of) series solutions centered at the origin of a binomial Horn system look, combinatorially?
• When is a holonomic -module?
• When is a regular holonomic -module?

We now summarize the main answers to these questions. We refer to [DMM06] for precise definitions, statements and proofs. We explicitly describe and classify all primary components of (in particular, all monomials that are present), their multiplicities, their behaviour with respect to the grading (which splits the components into toral (when they don't have any further homogeneities, as in the component at infinity of the Horn system, and Andean (when they admit a new homogeneity as in the component at infinity in the Appell system), and their holonomic rank. Moreover, we explicitly define three subspace arrangements associated to the Andean components (Andean arrangement), to the pairwise intersections of two components (primary arrangement) and to the rank-jumping parameters where the holonomic rank increases (jump arrangement), as the Zariski closure of parameters for which the corresponding piece in certain local cohomology modules is non zero, which account for non generic behaviour of the complex parameter .

Concretely,

• The dimension is finite exactly for not in the Andean arrangement.
• Given , denote by the submatrix of which consists only of the columns of for and denote by a sublattice of . We refer to the Introduction of [DMM06] for complete definitions and explanations, in particular for the definition of the multiplicities and the notion of toral associated sublattice. The generic (minimum) holonomic rank is , the sum being over all toral associated sublattices with , where is the volume of the convex hull of  and the origin, normalized so a lattice simplex in  has volume .
• The minimum rank is attained precisely when lies outside of an explicit affine subspace arrangement determined by certain local cohomology modules, containing the Andean arrangement.
• When the Horn system is regular holonomic and  is general, there are linearly independent solutions supported on (translates of) the -bounded classes (for the definition, see [DMM06, Subsection 1.6]), with hypergeometric recursions determining the coefficients.
• Only many Gamma series solutions have full support, where we denote the index of  in its saturation .
• Holonomicity is equivalent to finite dimension of the (local holomorphic) solutions spaces.
• Holonomicity is equivalent to regular holonomicity when is standard -graded—i.e., the row-span of  contains the vector . Conversely, if there exists a parameter for which is regular holonomic, then is -graded.

3.3. Main tools.

We briefly discuss the main tools in the proofs of the previous results. As we said in the introduction, these results are generalizations of the beautiful theory of -hypergeometric systems developed by Gelfand, Kapranov, and Zelevinsky in [GKZ89GKZ90], see also [Ado94SST00] for further developments of the basic theory. They defined the systems in terms of binomials and Euler operators, proved that they are holonomic, computed the holonomic rank for generic parameters, described the singular locus (in terms of sparse discriminants), and constructed bases of solutions for generic parameters in terms of Gamma series.

We also use the results in [DMM08], which give a precise description of the combinatorial and commutative algebra of primary components of binomial ideals in semigroup rings, based on the basic work of Eisenbud and Sturmfels on binomial ideals in [ES96]. In particular we have characterized the monomials in each primary component.

In characteristic zero, the primary decomposition of arbitrary binomial ideals is controled by the geometry and combinatorics of lattice point congruences, which also governs the -module theoretic properties of binomial -modules. In order to make this translation, we need to extend the results on Euler-Koszul homology from [MMW05], which functorially translate the commutative algebra of -graded primary decomposition into the -module setting. The Euler-Kozsul homology functor is used to pull apart the primary components of binomial ideals, thereby isolating the contribution of each to the solutions of the corresponding binomial -module. This allows us to integrate results from [GKZ89GKZ90Ado94Hot91SchW08DMS05]. In particular, we prove that for parameters  outside the (finite) primary subspace arrangement, the binomial -module decomposes as a direct sum over the toral primary components of .

4. Further examples: binomials and the shape of the solutions

The basic blocks in all these descriptions, besides the prime binomial ideals (i.e. the toric ideals), are the zero dimensional binomial ideals. We illustrate in this section how the information is assembled to give bases of series solutions for Horn binomial -modules. We refer the reader to [DMM06, Section t] for general definitions and results.

4.1. Finding polynomial solutions of square binomial ideals.

Let be a square matrix such that each column has at least one negative and one positive entry. Such an defines an infinite graph with vertices in the points and edges for all pairs such that is a column of . Consider the system of constant coefficient binomial differential operators , where the operator is associated to the -th column of as follows:

The number of bounded connected components of the graph corresponds precisely to the irreducible supports of polynomial solutions to . This number equals the dimension of the local quotient by at the origin. Moreover, let be the set of all the monomials occuring in a polynomial solution of . Then, equals precisely all monomials in the complement of a monomial ideal , which is the biggest monomial ideal (in the commutative ring generated by the partial derivatives) which lies inside .

Consider for instance the following matrix

The system  is defined by the operators

It has linearly independent polynomial solutions, with the following minimal supports:

The coefficients of these polynomial solutions are prescribed (up to constant) by the recurrence imposed by the binomial operators of the system. For instance, the bounded connected component gives rise to the quatrinomial solution of ;

 (4.1)

The set has monomials, precisely those monomials which do not belong to the monomial ideal , which is the biggest monomial ideal inside . Note that in this case, the matrix is invertible, so its "Gale dual" matrix would be empty.

4.2. Description of solutions to complete intersection Horn binomial D-modules.

Consider the matrices:

We can associate a Horn binomial -module to the columns of and a parameter vector :

where denote the binomials:

The variety has a primary component "at infinity" associated to the block decomposition of the matrix given by the following submatrices :

Note that , so there is only one associated primary component coming from this decomposition, with associated prime . Solutions to the binomial -module associated to can be constructed from the polynomial solutions to and solutions to the -hypergeometric system associated to the matrix , which, up to a rescaling in the homogeneities, equals the -hypergeometric system associated to the matrix , whose columns span .

For instance, consider the quatrinomial in (4.1). Let be any -hypergeometric function of -degree . Then, the following function is a solution of :

Note that all these solutions depend polynomially on .

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Alicia Dickenstein
Departamento de Matemática