BIVARIATE HYPERGEOMETRIC D-MODULES

ALICIA DICKENSTEIN, LAURA FELICIA MATUSEVICH, AND TIMUR SADYKOV

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.

1. Introduction

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, [10] and [12].

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:

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!} + ⋯

satisfies the differential equation:

z(1 − z)\frac{d^{2}f}{dz^{2}} + (c − (1 + a + b)z)\frac{df}{dz} − abf = 0.

Kummer went on to find all of the solutions of this equation (see [21]). He constructed twentyfour (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 [23]. For more historical details on hypergeometric functions, and a comprehensive treatment of their classical theory, see [27].

Both of these approaches have been tried for bivariate hypergeometric series. In his article [9], Erdélyi gives a complete set of solutions for the following system of two hypergeometric equations in two variables:

Alicia Dickenstein was partially supported by UBACYT X052 and ANPCYT 03-6568, Argentina.

Laura Felicia Matusevich was partially supported by a Sarah M. Hallam fellowship at UC Berkeley, and Liftoff fellowship from the Clay Mathematics Institute.

Timur Sadykov was partially supported by the Russian Ministry of Education, grant E 02-1-138.

(x(θ_{x} + θ_{y} + a)(θ_{x} + b) − θ_{x}(θ_{x} + θ_{y} + c − 1))f = 0 ,
(y(θ_{x} + θ_{y} + a)(θ_{y} + b′) − θ_{y}(θ_{x} + θ_{y} + c − 1))f = 0 ,

where θ_{x} = x\frac{∂}{∂x} and θ_{y} = y \frac{∂}{∂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, Erdélyi constructs more than 120 fully supported series solutions through contour integration. By a 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:

(x(2θ_{x} − θ_{y} + a′)(2θ_{x} − θ_{y} + a′ + 1) − (−θ_{x} + 2θ_{y} + a)θ_{x})f = 0 ,
(y(−θ_{x} + 2θ_{y} + a)(−θ_{x} + 2θ_{y} + a + 1) − (2θ_{x} − θ_{y} + a′)θ_{y})f = 0 .

This is the system of equations for Horn’s function G_{3}, and its holonomic rank is 4. Erdélyi 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 [13] about the Γ-series associated with lattices, and how they relate to Horn series. Notice that Γ-series as defined in [13] 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. [16, p. 296], 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.

2. Multivariate hypergeometric systems

In order to accommodate two different sets of variables, we denote by D_{n} the Weyl algebra with generators x_{1}, …, x_{n}, ∂_{x_{1}}, …, ∂_{x_{n}}, and by D_{m} the Weyl algebra whose generators are y_{1}, …, y_{m}, ∂_{y_{1}}, …, ∂_{y_{m}}. We set θ_{x_{j}} = x_{j}∂_{x_{j}} for 1 ≤ j ≤ n, and θ_{y_{i}} = y_{i}∂_{y_{i}}, for 1 ≤ i ≤ m. We also define θ_{x} = (θ_{x_{1}}, …, θ_{x_{n}}) and θ_{y} = (θ_{y_{1}}, …, θ_{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}) ∈ ℤ^{(n−m)×n} of full rank n − m whose first row is the vector (1, …, 1), and a matrix B ∈ ℤ^{n×m} = (b_{ji}) of full rank m such that A · B = 0. For 1 ≤ j ≤ m, set b_{j} = (b_{j1}, …, b_{jm}) ∈ ℤ^{m} the j-th row of B. The (positive) greatest common divisor of the maximal minors of the matrix B is denoted by g.

For i = 1, …, m, and a fixed parameter vector c = (c_{1}, …, c_{n}) ∈ ℂ^{n}, we let

P_{i} = ∏_{b_{ji}<0} ∏_{l=0}^{|b_{ji}|−1}(b_{j} · θ_{y} + c_{j} − l), (1)

Q_{i} = ∏_{b_{ji}>0} ∏_{l=0}^{b_{ji}−1}(b_{j} · θ_{y} + c_{j} − l), and (2)

H_{i} = Q_{i} − y_{i}P_{i}, (3)

where b_{j} · θ_{y} = ∑_{k=1}^{m} b_{jk}θ_{y_{k}}. The operators H_{i} are the Horn operators corresponding to the lattice L_{B} = {B ·z : z ∈ ℤ^{m}} and the parameter vector c. We call d_{i} = ∑_{b_{ij}>0} b_{ij} = − ∑_{b_{ij}<0} b_{ij} the order of the operator H_{i}.

Definition 2.1. The Horn system is the following left ideal of D_{m}:

Horn (B, c) = ⟨H_{1}, …, H_{m}⟩ ⊆ D_{m}.

Now denote by b^{(i)} the columns of the matrix B. Any vector u ∈ 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, …, m, we let:

T_{i} = ∂_{x}^{b_{+}^{(i)}} − ∂_{x}^{b_{−}^{(i)}},

here we use multi-index notation ∂_{x}^{v} = ∂_{x_{1}}^{v_{1}} ⋯ ∂_{x_{n}}^{v_{n}}. More generally, for any u ∈ L_{B}, set

T_{u} = ∂_{x}^{u_{+}} − ∂_{x}^{u_{−}}.

These are the lattice operators arising from L_{B}.

Definition 2.2. The lattice ideal arising from L_{B} is:

I_{B} = ⟨T_{u} : u ∈ L_{B}⟩ ⊆ ℂ[∂_{x_{1}}, …, ∂_{x_{n}}].

Recall that the toric ideal corresponding to A is:

I_{A} = ⟨T_{u} : u ∈ ker_{ℤ}(A)⟩ ⊆ ℂ[∂_{x_{1}}, …, ∂_{x_{n}}].

We will also denote:

I = ⟨T_{1}, …, T_{m}⟩ ⊆ ℂ[∂_{x_{1}}, …, ∂_{x_{n}}].

The ideal I is called a lattice basis ideal. Notice that, for m = 2, I is a complete intersection. This is not necessarily true if m > 2.

Lattice ideals and toric ideals have been extensively studied (see, for instance, [8], [28]). Lattice basis ideals were introduced in [18].

There is a natural system of differential equations arising from a toric ideal I_{A} and a parameter vector. This system, called the A-hypergeometric system with parameter A · c, is defined as:

H_{A}(A · c) = I_{A} + ⟨∑_{j=1}^{n} a_{ij}x_{j}∂_{x_{j}} − (A · c)_{i} : i = 1, …, n − m⟩ ⊆ D_{n}.

¿From now on we will use the notation ⟨A · θ − A · c⟩ to mean ⟨∑_{j=1}^{n} a_{ij}x_{j}∂_{x_{j}} − (A · c)_{i} : i = 1, …, n − m⟩.

A-hypergeometric systems were first defined by Gel′fand, Graev and Zelevinsky in [14], and their systematic analysis was started by Gel′fand, Kapranov and Zelevinsky (see, for instance, [15]). Saito, Sturmfels and Takayama have used Gröbner deformations in the Weyl algebra to study A-hypergeometric systems (see [26]). In this article, we will extend this approach to the case of Horn systems.

Gel′fand, Graev and Retakh have also considered the hypergeometric system associated with the lattice L_{B} = {B · z : z ∈ ℤ^{m}}, which is defined to be the left D_{n}-ideal:

I_{B} + ⟨A · θ − A · c⟩ ⊆ D_{n}.

We now introduce the D_{n}-ideal H_{B}(c), that is very closely related to the Horn system Horn (B, c):

H_{B}(c) = I + ⟨A · θ − A · c⟩ ⊆ D_{n}.

The results in Section 5 imply that, for generic c, there is a vector space isomorphism between the solution spaces of Horn (B, c) and H_{B}(c). Thus, we have two points of view to study Horn hypergeometric functions. We also call H_{B}(c) a Horn system, when the context is clear.

Remark 2.3. We have defined the Horn operators using falling factorials because this formulation will make clearer the relationship between Horn (B, c) and H_{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.

It is a well known result of Adolphson [1] that, for generic parameters A · 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 [24]).

Definition 2.4. In the case that m = 2, we set

ν_{ij} = \{ \begin{matrix}min(|b_{i1}b_{j2}|, |b_{j1}b_{i2}|), & if b_{i}, b_{j} are in the interior of opposite quadrants of ℤ^{2}, \\ 0 & otherwise, \end{matrix}

for 1 ≤ i, j ≤ n. The number ν_{ij} is called the index associated to b_{i} and b_{j}.

The following is the main result in this article, which follows from Theorems 8.8, 9.5 and 10.7.

Theorem 2.5. Let B be an n × 2 integer matrix of full rank such that its rows b_{1}, …, b_{n} satisfy b_{1} + ⋯ + b_{n} = 0. If c ∈ ℂ^{n} is a generic parameter vector, then the ideals Horn (B, c) and H_{B}(c) are regular holonomic. Moreover,

rank (H_{B}(c)) = rank (Horn (B, c)) = d_{1}d_{2} − ∑ ν_{ij} ,

where the sum runs over all pairs of linearly dependent rows b_{i}, b_{j} of B lying in opposite open quadrants of ℤ^{2}.

We can also give an explicit basis for the solution space of Horn (B, c) (and of H_{B}(c)) (Theorem 9.6), and compute the exact dimension of the subspace of Puiseux polynomial solutions (Theorem 6.5).

3. Some observations about Horn systems

The Horn system Horn (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 (B, c), since this system is homogeneous. Moreover, as we will see in Section 5, all the solutions of the A-hypergeometric system H_{A}(A · c) are solutions of H_{B}(c), and these can be transformed into solutions of Horn (B, c) (see Corollary 5.2), so that, under the assumptions that B is n × m of full rank m, n > m, with all column sums equal to zero, Horn (B, c) always has nonzero solutions, since H_{A}(A · c) always has nonzero solutions (its solution space has dimension at least deg(I_{A}) = vol (A), see [26, Theorem 3.5.1]).

It is easy to understand how the Horn system Horn (B, c) changes if we choose a new parameter vector c′, as long as A · c′ = A · c. As a matter of fact, if c = c′ + B · z, for some z ∈ ℂ^{m}, then it is easy to see that f(y) is a solution of Horn (B, c′) if and only if y^{z}f(y) is a solution of Horn (B, c). Notice also that the system H_{B}(c) depends only on A · c, so that H_{B}(c) = H_{B}(c′) if A · c = A · c′.

A change in A·c can, instead, alter the solution space of Horn (B, c) (and H_{B}(c)) in dramatic ways. For instance, it could become infinite-dimensional, as the following example shows.

Example 3.1. The Horn system defined by the operators

(4)  (θ_{y_{1}} + θ_{y_{2}} + c_{1})θ_{y_{i}} − y_{i}(θ_{y_{1}} + θ_{y_{2}} + c_{2})(θ_{y_{1}} + θ_{y_{2}} + c_{3}), i = 1, 2

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 θ_{y_{1}} + θ_{y_{2}} + c_{1} can be factored out of each of the operators in (4), it follows that any function which is annihilated by θ_{y_{1}} + θ_{y_{2}} + c_{1} is a solution to (4). Thus for any smooth univariate function u the product y_{2}^{−c_{1}}u(y_{1}/y_{2}) satisfies (4).

Notice that for generic values of the parameters c_{1}, c_{2}, c_{3} the system (4) 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.

We could also ask what happens if we choose another matrix B′ such that A · B′ = 0. Even if g = g′ = 1, so that B and B′ are two Gale duals of A, the associated Horn systems could have different holonomic rank, as we see in Example 3.2. The systematic analysis of this question, in the case when m = 2 is one of the main objectives of this article.

Example 3.2. We choose:

A = ( \begin{matrix}1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \end{matrix} ) , B = ( \begin{matrix}1 & 0 \\ −2 & 1 \\ 1 & −2 \\ 0 & 1 \end{matrix} ) , B′ = ( \begin{matrix}1 & 2 \\ −2 & −3 \\ 1 & 0 \\ 0 & 1 \end{matrix} ) .

Then, if c is a generic parameter vector, rank (Horn (B, c)) = 4, and rank (Horn (B′, c)) = 6, as a consequence of Theorem 2.5. This can be verified for specific values of c using the computer algebra system Macaulay 2 [17]. However, by Theorem 5.3, these two hypergeometric systems share all fully supported solutions.

Notice that the definition of Horn (B, c) makes sense even if B is a square matrix, or if the rows of B don’t add up to zero, or even if 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 B is n × m of full rank m, m < n, and the rows of B add up to zero. Many of the examples will also concern Horn systems with n = m. We remark that if B is square and nonsingular, then H_{B}(c) is a system of differential equations with constant coefficients, not depending on c.

4. Preliminaries on codimension 2 binomial ideals

In this section we collect some results about lattice ideals and lattice basis ideals that will be necessary to study Horn systems. Although this section is about commutative algebra, our indeterminates will be called ∂_{1}, …, ∂_{n} for consistency with the notation for differential equations.

Recall that B = (b_{ji}) is an n × m integer matrix of full rank m with all column sums equal to zero. The following ideal is called a lattice ideal:

I_{B} = ⟨∂^{u_{+}} − ∂^{u_{−}} : u = u_{+} − u_{−} ∈ L_{B}⟩ ⊂ ℂ[∂_{1}, …, ∂_{n}],

where L_{B} = {B·z : z ∈ ℤ^{m}} is the rank-m lattice spanned by the columns of B. For the purpose of this section, we could use any field of characteristic 0 instead of ℂ, but later on, when we talk about complex holomorphic solutions of differential equations, we will need our field to be the complex numbers. We let A be any (n − m) × n integer matrix such that A · B = 0. Then the saturation of L_{B} is the lattice L = ker_{ℤ}(A). Notice that the order of the group L/L_{B} is g, the positive greatest common divisor of the maximal minors of B.

The ideal I_{B} is homogeneous with respect to the usual ℤ-grading and hence defines a subscheme X_{B} of ℙ^{n−1}. Moreover, the ideal I_{B} is always radical and X_{B} is the equidimensional union of g = |L/L_{B}| torus translates of the toric variety X_{A} defined by the reduced scheme associated to L as above. This is deduced from [8] since (I_{B} : ⟨∂_{1}, …, ∂_{n}⟩^{∞}) = I_{B}, that is, no component of X_{B} is contained in a coordinate hyperplane.

These torus translates can be described in terms of the order g group G_{B} of all partial characters ρ : L → ℂ^{∗} which extend the trivial character 1 : L_{B} → ℂ^{∗}, i.e., ρ satisfying ρ(ℓ + ℓ′) = ρ(ℓ)ρ(ℓ′), ∀ ℓ, ℓ′ ∈ L and ρ(ℓ) = 1, ∀ ℓ ∈ L_{B}.

Example 4.1. We illustrate the previous decomposition in an example before writing it down in general. Let

B = ( \begin{matrix}−1 & 2 \\ 0 & −3 \\ 3 & 0 \\ −2 & 1 \end{matrix} ) , A = ( \begin{matrix}1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 3 \end{matrix} ) .

In this case g = 3. The scheme X_{A} is the twisted cubic, that is, the closure of the torus orbit of the point p_{0} = (1 : 1 : 1 : 1) ∈ ℙ^{3} under the torus action:

(5)  λ · (∂_{1} : ∂_{2} : ∂_{3} : ∂_{4}) = (λ^{0}∂_{1} : λ^{1}∂_{2} : λ^{2}∂_{3} : λ^{3}∂_{4}), λ ∈ ℂ^{∗}.

The group G_{B} has order 3 and is isomorphic to the group of cubic roots of unity {1, ω, ω^{2}}, where ω = e^{\frac{2πi}{3}} . Set p_{1} = (1 : 1 : ω : 1), p_{2} = (1 : 1 : ω^{2} : 1) and denote by X_{0}, X_{1} and X_{2} the respective closure of the torus orbit under the action (5) of p_{0}, p_{1} and p_{2}. In particular, X_{0} = X_{A}. Then

X_{B} = X_{0} ∪ X_{1} ∪ X_{2}

and X_{i} is the image of X_{0} under the coordinatewise multiplication by p_{i}, i = 1, 2. Note that

X_{i} = {(∂_{1} : ⋯: ∂_{4}) : ∂_{1}∂_{3} − ω^{i}∂_{2}^{2} = ∂_{3}^{2} − ω^{2i}∂_{2}∂_{4} = ∂_{2}∂_{3} − ω^{i}∂_{1}∂_{4} = 0}

so that the equations defining X_{i} are “translations” of the equations for X_{0} = X_{A}.

This can be phrased in general as follows: Given ρ ∈ G_{B}, let X_{ρ} denote zero scheme of the ideal:

I_{ρ} = ⟨∂^{u_{+}} − ρ(u)∂^{u_{−}} : u = u_{+} − u_{−} ∈ L⟩.

Then the ideals I_{ρ} are prime, their intersection gives I_{B} and X_{B} = ∪_{ρ∈G_{B}}X_{ρ}. We refer to [8] for a proof of these facts.

Consider now the case m = 2 and recall that the lattice basis ideal associated to B is the ideal

I = ⟨∂^{u_{+}} − ∂^{u_{−}} : u is a column of B⟩.

Its zero set consists of the union of X_{B} with components that lie inside coordinate hyperplanes. The following proposition, whose proof can be found in [6], gives the precise primary decomposition of the ideal I. Denote b_{1}, …, b_{n} ∈ ℤ^{2} the row vectors of B. Let ν_{ij} be the index associated to b_{i} and b_{j} as in Definition 2.4.

Proposition 4.2. The ideal I has the following primary decomposition:

I = ( ∩_{ρ∈G_{B}} I_{ρ}) ∩ ( ∩_{ν_{ij}>0} I_{ij})

where \sqrt{I_{ij}} = ⟨∂_{i}, ∂_{j}⟩, and the multiplicity of each I_{ij} is ν_{ij}, in the sense that

dim_{K}(ℂ[∂_{1}, …, ∂_{n}]/I_{ij})_{⟨∂_{1},..., ˆ∂_{i},..., ˆ∂_{j},...,∂_{n}⟩} = ν_{ij},

where K = ℂ(∂_{1}, …, ˆ∂_{i}, …, ˆ∂_{j}, …, ∂_{n}).

We then have

Corollary 4.3. For d_{1}, d_{2} the degrees of the generators of I,

(6)  d_{1} · d_{2} − ∑_{b_{i},b_{j} dependent} ν_{ij} = g · vol (A) + ∑_{b_{i},b_{j} independent} ν_{ij} ,

where the first summation runs over linearly dependent pairs b_{i}, b_{j} of rows of B that lie in opposite open quadrants of ℤ^{2}, and the second summation runs over linearly independent such pairs.

Proof. The degree of the complete intersection I is d_{1}d_{2}. By Proposition 4.2, this number equals

g · deg(I_{A}) + ∑ ν_{ij} ,

where the sum runs over all pairs of rows of 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 [28, Theorem 4.16]. □

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.

Proposition 4.4. Let B ∈ ℤ^{n×2} of rank 2, with rows b_{1}, …, b_{n}, that add up to zero, and I_{B}, I, the lattice and lattice basis ideals associated to B. For each 1 ≤ i, j ≤ n, ν_{ij} is as in Definition 2.4. Set

α_{i} = \{ \begin{matrix}max_{j} ν_{ij} & if b_{i1} > 0, \\ 0 & otherwise. \end{matrix}

Then

∂^{α}I_{B} ⊆ I in ℂ[∂_{1}, …, ∂_{n}].

Proof. By Proposition 4.2, it is enough to prove that ∂^{α} ∈ ∩_{ν_{ij}>0}I_{ij}. Assume that ν_{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 α_{i} ≥ ν_{ij}. We will be done if we show that ∂_{i}^{ν_{ij}} ∈ I_{ij}. To do this, let Ĩ_{ij} be the localization of I_{ij} at ⟨∂_{1}, …, ˆ∂_{i}, …, ˆ∂_{j}, …, ∂_{n}⟩ so that Ĩ_{ij} is an artinian ideal of multiplicity ν_{ij} in K[∂_{i}, ∂_{j}], where K = ℂ(∂_{1}, …, ˆ∂_{i}, …, ˆ∂_{j}, …, ∂_{n}). Notice that, since #{1, ∂_{i}, …, ∂_{i}^{ν_{ij}}} = ν_{ij} + 1, these monomials must be linearly dependent modulo Ĩ_{ij}, so we can find g_{0}, …, g_{ν_{ij}} ∈ K such that

g_{0} + g_{1}∂_{i} + ⋯ + g_{ν_{ij}}∂_{i}^{ν_{ij}} ∈ Ĩ_{ij}.

But the radical of Ĩ_{ij} is ⟨∂_{i}, ∂_{j}⟩, so that g_{0} = 0. Let l = min_{1≤k≤ν_{ij}}{g_{k} ≠ 0}. Then, clearing denominators, we can find polynomials f_{l}, …, f_{ν_{ij}} not involving the variables ∂_{i}, ∂_{j}, f_{l} ≠ 0, such that

∂_{i}^{l}(f_{l} + ⋯ + f_{ν_{ij}}∂_{i}^{ν_{ij}−l}) ∈ I_{ij}.

Now, since I_{ij} is primary to ⟨∂_{i}, ∂_{j}⟩, and no power of f_{l} + ⋯ + f_{ν_{ij}}∂_{i}^{ν_{ij}−l} belongs to ⟨∂_{i}, ∂_{j}⟩, then ∂_{i}^{l} must belong to I_{ij}. Since l ≤ ν_{ij}, we are done. □

It is an interesting fact that the multiplicities of some of the components of I do not go down under Gröbner deformation. Given w ∈ ℤ^{n}, and f = ∑ f_{α}x^{α} a homogeneous polynomial in ℂ[∂_{1}, …, ∂_{n}], let

in_{w}(f) = ∑_{w·α maximal over f_{α}≠0} f_{α}x^{α}

and define

in_{w}(I) = ⟨in_{w}(f) : f ∈ I \ {0}⟩.

The ideal in_{w}(I) is called the initial ideal of I with respect to the weight vector w. It is a monomial ideal if w is generic (see [5] and [7, Chapter 15] for more on initial ideals, especially how to compute them).

Lemma 4.5. Let b_{k} and b_{l} be two linearly dependent rows of B lying in opposite open quadrants of ℤ^{2}. If w is a generic weight vector, then the multiplicity of the ideal ⟨∂_{k}, ∂_{l}⟩ as an associated prime of in_{w}(I) is at least the index ν_{kl}.

As a matter of fact, the multiplicities in the previous lemma will be preserved under taking initial ideals (see Corollary 9.7).

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 ≤ r (so that s ≥ q, since ps = qr). The ideal I is generated by binomials ∂_{k}^{p}m_{1} − ∂_{l}^{q}m_{2} and ∂_{k}^{r}m_{3} − ∂_{l}^{s}m_{4}, where m_{1}, …, m_{4} are monomials that do not contain the variables ∂_{k}, ∂_{l}. We need only consider two cases:

(i) The leading monomials of the generators of I are ∂_{k}^{p}m_{1} and ∂_{s}^{s}m_{4}.

(ii) The leading monomials of the generators of I are ∂_{k}^{p}m_{1} and ∂_{k}^{r}m_{3}.

In case (i), it is clear that any other initial monomial (obtained by taking S-pairs) will contain a power of ∂_{k} greater than or equal to p or a power of ∂_{l} greater than or equal to s. This implies that ⟨∂_{k}, ∂_{l}⟩ is an associated prime of in_{w}(I), with multiplicity at least ps = ν_{kl}.

Now suppose we are in case (ii), that is, the initial monomials of our two binomials are ∂_{k}^{p}m_{1} and ∂_{k}^{r}m_{3}. Choose an integer t such that r ≤ tp.

The S-pair between ∂_{k}^{p}m_{1} − ∂_{l}^{q}m_{2} and ∂_{k}^{r}m_{3} − ∂_{l}^{s}m_{4} is:

∂_{l}^{s}m_{1}m_{4} − ∂_{k}^{r−p}∂_{l}^{q}m_{2}m_{3}.

If the first monomial is the leading term, as in case (i), we see that every initial monomial has a factor of ∂_{k}^{p} or ∂_{l}^{s}, and we are done. Otherwise, we take the S-pair of ∂_{k}^{p}m_{1} − ∂_{l}^{q}m_{2} and the previous binomial. We obtain, up to a sign,

m_{1}^{2}m_{4}∂_{l}^{s} − m_{2}^{2}m_{3}∂_{k}^{r−2p}∂_{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}∂_{k}^{r−(t−1)p}∂_{l}^{(t−1)q} is the leading term of the S-pair

m_{2}^{t−1}m_{3}∂_{k}^{r−(t−1)p}∂_{l}^{(t−1)q} − m_{1}^{t−1}m_{4}∂_{l}^{s}.

We consider the two possible situations: r = tp and r < tp. If r = tp, tq = s and the S-pair with ∂_{k}^{p}m_{1} − ∂_{l}^{q}m_{2} is:

∂_{l}^{s}m_{1}^{t}m_{4} − ∂_{l}^{tq}m_{2}^{t}m_{3} = ∂_{l}^{s}(m_{1}^{t}m_{4} − m_{2}^{t}m_{3}).

We want to show that this binomial is nonzero. But if m_{1}^{t}m_{4} = m_{2}^{t}m_{3}, then m_{2}^{t}(∂_{k}^{r}m_{3} − ∂_{l}^{s}m_{4}) = m_{4}(∂_{k}^{tp}m_{1}^{t} − ∂_{l}^{tq}m_{2}^{t}), and this implies that the zero set of I is at least one dimensional, contradicting that I is a complete intersection.

Now suppose that r < tp. Then, r − (t − 1)p < p and taking S pair again with ∂_{k}^{p}m_{1} − ∂_{l}^{q}m_{2}, we get:

∂_{k}^{tp−r}∂_{l}^{s}m_{1}^{t}m_{4} − ∂_{l}^{tq}m_{2}^{t}m_{3}.

Since s < tq, no matter what the leading term is, it will have a factor of ∂_{l}^{s}. As before, we conclude that ⟨∂_{k}, ∂_{l}⟩ is associated to in_{w}(I), with multiplicity at least ps = ν_{kl}. This finishes the proof. □

5. A-hypergeometric solutions of the Horn system

In this section we study the solutions of the Horn system Horn (B, c) that arise from the A-hypergeometric system H_{A}(A · c). Here, we do not use the assumption that m = 2. Recall that B = (b_{ji}) is an rank m integer n×m matrix whose rows add up to zero, and whose columns are denoted b^{(1)}, …, b^{(m)} and let A = (a_{ij}) be any rank (n − m) integer (n − m) × n matrix such that A · B = 0. Here we assume that n > m.

Consider the surjective map

x^{B} : (ℂ^{∗})^{n} → (ℂ^{∗})^{m},
x ↦ (∏_{j=1}^{n} x_{j}^{b_{j1}}, …, ∏_{j=1}^{n} x_{j}^{b_{jm}}) = (x^{b^{(1)}}, …, x^{b^{(m)}}).

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 2.

Lemma 5.1. Let U ⊆ (ℂ^{∗})^{n} be a simply connected open set and let V = x^{B}(U). We choose U small enough so that V is also simply connected. Given a holomorphic function ψ ∈ O(V), call φ = x^{c}ψ(x^{B}). Then

i. ( ∑_{j=1}^{n} a_{kj}x_{j}∂_{x_{j}})(φ) = (A · c)_{k}φ, for k = 1, …, n − m.

ii. T_{i}(φ) = 0 for i = 1, …, m if and only if H_{i}(ψ) = 0 for i = 1, …, m.

iii. Moreover, for any u = B · z ∈ ℤ^{n} ∈ L_{B}, and

H_{u} = ∏_{u_{j}>0} ∏_{l=0}^{u_{j}−1}(b_{j} · θ_{y} + c_{j} − l) − y^{z} ∏_{u_{j}<0} ∏_{l=0}^{|u_{j}|−1}(b_{j} · θ_{y} + c_{j} − l),

we have T_{u}(φ) = 0 if and only if H_{u}(ψ) = 0.

Proof. The verifications of the three assertions are very similar. The main ingredients are the following identities:

(7)  θ_{x_{i}}x^{c} = x^{c}(θ_{x_{i}} + c_{i}), (in D_{n}),

(8)  θ_{x_{i}}(ψ(x^{B}))(x) = [(b_{i} · θ_{y})ψ](x^{B}),

which are easily checked. Let us prove (ii). Call T̃_{i} = ∏_{b_{ji}>0} x_{j}^{b_{ji}}T_{i}. We have:

(9)  T̃_{i} = ∏_{b_{ji}>0} x_{j}^{b_{ji}} ∏_{b_{ji}>0} ∂_{x_{j}}^{b_{ji}} − (x^{B})_{i} ∏_{b_{ji}<0} x_{j}^{−b_{ji}} ∏_{b_{ji}<0} ∂_{x_{j}}^{−b_{ji}}.

Recall that (x^{B})_{i} = ∏_{j=1}^{n} x_{j}^{b_{ji}}. Using the identity:

x^{α}∂_{x}^{α} = ∏_{j=1}^{n} ∏_{l=0}^{α_{j}−1}(θ_{x_{j}} − l) ,

equation (9) is transformed into:

T̃_{i} = ∏_{b_{ji}>0} ∏_{l=0}^{b_{ji}−1}(θ_{x_{j}} − l) − (x^{B})_{i} ∏_{b_{ji}<0} ∏_{l=0}^{−b_{ji}−1}(θ_{x_{j}} − l)

Using (7),

T̃_{i}(φ) = T̃_{i}(x^{c}ψ(x^{B}))

= x^{c}( ∏_{l=0}^{b_{ji}−1}(θ_{x_{j}} + c_{j} − l) − (x^{B})_{i} ∏_{b_{ji}<0} ∏_{l=0}^{−b_{ji}−1}(θ_{x_{j}} + c_{j} − l))(ψ(x^{B})).

Now (8) implies that

T̃_{i}(φ) = x^{c}( ∏_{l=0}^{b_{ji}−1}(b_{j} · θ_{y}(ψ) + c_{j} − l)−

− (x^{B})_{i} ∏_{b_{ji}<0} ∏_{l=0}^{−b_{ji}−1}(b_{j} · θ_{y}(ψ) + c_{j} − l))(x^{B})

= x^{c}H_{i}(ψ)((x^{B}) .

This shows that T̃_{i}(φ) is identically zero if and only if H_{i}(ψ)(x^{B}) = 0 for all x ∈ U. This is equivalent to H_{i}(ψ) vanishing identically on V. Since T_{i}φ = 0 if and only if T̃_{i}φ = 0, we obtain the desired result. □

Parts (i) and (ii) of Lemma 5.1 have the following consequence.

Corollary 5.2. The map

{Holomorphic solutions of Horn (B, c) on V } ⟶ {Holomorphic solutions of H_{B}(c) on U}
ψ ⟼ x^{c}ψ(x^{B})

is a vector space isomorphism, that takes Puiseux polynomials to Puiseux polynomials.

Finally, we can use the solutions of H_{A}(A · c) to construct solutions of H_{B}(c) (and thus of Horn (B, c)). We refer to [26, Section 3] 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.

Theorem 5.3. Given a generic parameter vector c, and {ϕ^{k} : k = 1, …, vol (A)} a canonical basis for the space of solutions of the A-hypergeometric system H_{A}(A · c), there exist linearly independent, fully supported solutions with disjoint supports

{ψ_{l}^{k} : k = 1, …, vol (A), l = 1, …, g}

of Horn (B, c) such that

ϕ^{k} = x^{c} ∑_{l=1}^{g} ψ_{l}^{k}(x^{B}) , for all k = 1, …, vol (A) .

Moreover, no (non trivial) linear combination of the functions ψ_{l}^{k} is ever a Puiseux polynomial. This natural decomposition holds as well for canonical series solutions with logarithms.

Proof. By [26, Section 2.5] and [25, Proposition 5.2], a canonical series solution ϕ of the Ahypergeometric system H_{A}(A · c) is of the form

(10)  ϕ = x^{α} ∑ λ_{u,v}x^{u} log(x^{v}),

with A · α = A · c, and v, u ∈ L = ker_{ℤ}(A). We show that ϕ can be decomposed as a sum of g solutions ψ_{1}, …, ψ_{g} of H_{B}(c) such that, if ψ_{j}, ψ_{l} are nonzero, then they have disjoint supports. Observe that, if u, v ∈ L, then:

(11)  ((A · θ)_{j} − (A · c)_{j})(x^{u+α} log(x^{v})) = 0, and

(12)  ∂_{i}(x^{u+α} log(x^{v})) = (u + α)_{i} x^{u+α−e_{i}} log(x^{v}) + v_{i}x^{u+α−e_{i}}.

Consider the lattice L_{B} ⊆ ℤ^{n} generated by the columns of 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, …, g} be a system of representatives for L/L_{B}. Define

ψ_{l} = x^{α} ∑_{u≡u_{l}mod L} λ_{u,v}x^{u} log(x^{v}).

Clearly, ϕ = ψ_{1} + ⋯ + ψ_{g}, and the summands have pairwise disjoint support. By (11), each ψ_{l} is a solution of the system of homogeneities ⟨A · θ − A · c⟩. Now we need to check that each ψ_{l} is a solution of the binomial operators T_{1}, …, T_{m} given by the columns of B. Consider T_{j} = ∂^{b_{+}^{(j)}} − ∂^{b_{−}^{(j)}}. Certainly T_{j}ϕ = 0. We apply the operator T_{j} to ϕ = ψ_{1} + ⋯ + ψ_{g}, and observe that terms coming from T_{j} applied to ψ_{l} cannot cancel with terms coming from ∂^{b_{+}^{(j)}} nor from ∂^{b_{−}^{(j)}} applied to ψ_{l′} if l ≠ l′. This is because the exponents of the monomials appearing in (∂^{b_{+}^{(j)}})(ψ_{l}), for instance, are b_{+}^{(j)}-translates of the exponents of the monomials from ψ_{l} by (12), and b_{+}^{(j)} − b_{−}^{(j)} ∈ L_{B}. The lack of cancellation now follows from the fact that the supports of ψ_{l} and ψ_{l′} are not congruent modulo L_{B} by construction.

Now, if we have a canonical basis {ϕ^{k}, k = 1, …, vol(A)} for the space of solutions of H_{A}(A·c) for generic c ∈ ℂ, they are of the form (10)

ϕ^{k} = x^{α_{k}} ∑_{u∈L∩C_{k}} λ_{u,v}x^{u},

for different exponents α_{k} with respect to a generic weight vector, and u ranging over all lattice points in a full dimensional pointed cone C_{k}. Notice that, since c is generic, no pair of the exponents α_{k} can differ by an integer vector. Decompose each ϕ^{k} = ϕ_{1}^{k} + ⋯ + ϕ_{g}^{k} as above. Note that all ϕ_{l}^{k} are non zero; in fact, the convex hull of all the supports is full dimensional. Moreover, the collection ϕ_{l}^{k}, k = 1, …, vol(A), l = 1, …, g is linearly independent since the supports are disjoint. By Lemma 5.1, each ϕ_{l}^{k} is of the form x^{c}ψ_{l}^{k}(x^{B}), where ψ_{l}^{k} is a solution of Horn (B, c). Clearly, no (non trivial) linear combination of the functions ψ_{l}^{k} is ever a Puiseux polynomial; in particular, they are linearly independent. □

6. Puiseux polynomial solutions of the Horn system

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 B[i, j] be the square submatrix of 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 (B, c) is to reduce this computation to the case when B is a 2 × 2 matrix.

Lemma 6.1. For a generic parameter vector c,

rank_{p}(Horn (B, c)) = ∑_{i<j} rank_{p}(Horn (B[i, j], c[i, j])).

Proof. We call the support S of a solution of Horn (B, c) irreducible if there exists no other solution whose support is a proper nonempty subset of S. Let f(y) be a series solution to Horn (B, c) with irreducible support S and let s_{0} ∈ S. It follows by Theorem 1.3 in [24] that if the monomial y^{s_{0}} is not present in the series f(y) then for no s ∈ S can y^{s} be present in f(y). This implies that irreducible supports are disjoint. Indeed, if S_{1} and S_{2} are irreducible and s_{0} ∈ S_{1} ∩ S_{2} then there exist solutions f_{1} (respectively f_{2}) of Horn (B, c) supported in S_{1} (respectively S_{2}) such that f_{1} − f_{2} does not contain y^{s_{0}}. But then, since y^{s_{0}} does not appear in f_{1} − f_{2}, no monomial in S_{2} can appear in f_{1} − f_{2}, and hence S_{1}\S_{2} supports a solution of Horn (B, c). This contradicts the fact that S_{1} was irreducible.

Any Puiseux polynomial solution of Horn (B, c) can be written as a linear combination of polynomial solutions with irreducible supports. Since Puiseux polynomials with disjoint supports are linearly independent, it is sufficient to count irreducible supports in order to determine rank_{p}(Horn (B, c)).

Remember that the equations of the Horn system translate into recurrence relations for the coefficients of any of its power series solutions. We refer to [24] for a detailed study of these recurrences. They imply that any coefficient in a solution of a Horn system is given by a nonzero multiple of any of its adjacent coefficients, as long as none of the polynomials P_{i}, Q_{i} vanish at the corresponding exponent. This yields that the support of a solution must be “bounded” by the zeros of these polynomials in the following sense. The exponent of a monomial in a solution must lie in the zero locus of at least one of the polynomials P_{i}, Q_{i}, provided that some of the adjacent exponents are not present in the polynomial solution (See Theorem 1.3 in [24]).

Let S be the support of a Puiseux solution of Horn (B, c). If S is irreducible, then for a generic vector c the set S cannot meet more than two lines of the form b_{j} · θ_{y} + c_{j} − l = 0 corresponding to different parameters c_{j}. If it only meets one such line then by Theorem 1.3 in [24] the set S cannot be finite (in fact, its convex hull is a half-plane in this case). If S meets two lines of the above form then all the other lines can be removed from the picture without affecting the supports (but not the coefficients) of the Puiseux polynomial solutions which are generated by this specific pair of lines. This implies the desired result. □

Now our goal is to compute rank_{p}(Horn (B[i, j], c[i, j])). The first step is to eliminate the cases when this rank is zero.

Lemma 6.2. The system Horn (B[i, j], c[i, j]) has non-zero Puiseux polynomial solutions only if b_{i} and b_{j} are linearly independent in opposite open quadrants of ℤ^{2}, or for some special values of c_{i}, c_{j} when b_{i}, b_{j} are linearly dependent and opposite. The corresponding Puiseux polynomial solutions of H_{B[i,j]}(c[i, j]) are Taylor polynomials, that is, polynomials with natural number exponents.

Proof. Corollary 5.2 gives a vector space isomorphism between the solution spaces of the hypergeometric systems Horn (B[i, j], c[i, j]) and H_{B[i,j]}(c[i, j]) that takes Puiseux polynomials to Puiseux polynomials. Thus it is enough to investigate the Puiseux polynomial solutions of H_{B[i,j]}(c[i, j]). If b_{i} and b_{j} do not lie in the interior of opposite open quadrants, one of the operators in H_{B[i,j]}(c[i, j]) is of the form ∂^{α} − 1 for some α ∈ ℕ^{2}. It is clear that such an operator cannot have a Puiseux polynomial solution.

Now assume that b_{i} and b_{j} lie in the interior of opposite quadrants. Let us prove the statement about Taylor polynomials. We may without loss of generality assume that b_{i1} > 0. If b_{i2} < 0, then the change of variables ỹ_{1} = y_{1}, ỹ_{2} = 1/y_{2}, transforms Horn (B[i, j], c[i, j])) into a Horn system given by a 2 × 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_{B[i,j]}(c[i, j]) = ⟨∂_{i}^{b_{i1}} − ∂_{j}^{−b_{j1}}, ∂_{i}^{b_{i2}} − ∂_{j}^{−b_{j2}}⟩,

and this is an ideal in the Weyl algebra with generators x_{i},x_{j}, ∂_{i},∂_{j}.

Let us show that any Puiseux polynomial solution f of H_{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}) ∈ supp (f) such that Re u_{0} = min{Re u : (u, v) ∈ supp (f)\ℕ^{2}}. Then (∂_{i}^{b_{i1}} − ∂_{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} ∉ 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 ∈ ℕ for all (u, v) ∈ supp (f)\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 (∂_{i}^{b_{i1}} − ∂_{j}^{−b_{j1}})f = 0. But now v ∈ ℕ for all (u, v) ∈ supp (f)\ℕ^{2}. We conclude that supp (f) ⊂ ℕ^{2}.

Finally, let us show that if b_{i} and b_{j} are linearly dependent, then Horn (B[i, j], c[i, j]) only has the identically zero solution as long as c is generic. Using the change of variables ξ_{1} = y_{1}^{1/b_{i1}}, ξ_{2} = y_{2}^{1/b_{i2}}, we transform the operator b_{i} · θ_{y} to the operator θ_{ξ_{1}} + θ_{ξ_{2}}. By Lemma 10.4 (to be proved in Section 10) there exists a nonzero polynomial in y_{1}, y_{2} which lies in the ideal Horn (B[i, j], c[i, j]). Thus the only holomorphic solution to the system is the zero function. □

Example 6.3. Let us construct the Puiseux polynomial solutions to the system of equations Horn (B, 0), where

B = ( \begin{matrix}4 & 5 \\ −3 & −5 \end{matrix} ) .

The system H_{B}(0) is defined by the operators

(13)  \frac{∂^{4}}{∂x_{1}^{4}} − \frac{∂^{3}}{∂x_{2}^{3}} ,  \frac{∂^{5}}{∂x_{1}^{5}} − \frac{∂^{5}}{∂x_{2}^{5}} .

Notice that we may use the parameter 0 without loss of generality. The solutions of H_{B}(c) are exactly the same as those of H_{B}(0), and in the case of Horn (B, c), the only effect is a translation of the supports of the solutions.

The supports of the polynomial solutions to (13) are displayed in Figure 1. 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 B.

Figure 1. The supports of the 15 polynomial solutions to (13)

The polynomial solutions to (13) are given by

1, x_{1}, x_{1}^{2}, x_{1}^{3}, x_{2}, x_{1}x_{2}, x_{1}^{2}x_{2}, x_{1}^{3}x_{2}, x_{2}^{2}, x_{1}x_{2}^{2}, x_{1}^{2}x_{2}^{2}, x_{1}^{3}x_{2}^{2}, x_{1}^{4} + 4x_{2}^{3},

x_{1}^{4}x_{2} + x_{2}^{4}, 5x_{1}^{4}x_{2}^{2} + 2x_{1}^{5} + 2x_{2}^{5} + 40x_{1}x_{2}^{3}.

Now let us unravel our isomorphism of solution spaces to obtain the corresponding solutions of Horn (B, 0). As in the proof of the previous lemma, if ψ = ∑ ψ_{α}y^{α} is a Puiseux polynomial solution of Horn (B, 0), and ψ_{α} ≠ 0, then \binom{u}{v} = B · α ∈ ℕ^{2}. But then

α = B^{−1} · \binom{u}{v} = ( \begin{matrix}1 & 1 \\ −3/5 & −4/5 \end{matrix} ) · \binom{u}{v}.

This implies that α_{1} is a natural number, and α_{2} ∈ (−1/5)ℕ . Moreover B · α ≥ 0. Thus, in order to find the irreducible supports of the Puiseux polynomial solutions of Horn (B, 0), we need to draw the region B · α ≥ 0, plot the points α ∈ ℕ × (−1/5)ℕ, and connect those points with horizontal and vertical moves. This is done in Figure 2. 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 (B, 0) are as follows:

1, y_{1}y_{2}^{−3/5}, y_{1}^{2}y_{2}^{−6/5}, y_{1}^{3}y_{2}^{−9/5}, y_{1}y_{2}^{−4/5}, y_{1}^{2}y_{2}^{−7/5}, y_{1}^{3}y_{2}^{−2}, y_{1}^{4}y_{2}^{−13/5}, y_{1}^{2}y_{2}^{−8/5},

y_{1}^{3}y_{2}^{−11/5}, y_{1}^{4}y_{2}^{−14/5}, y_{1}^{5}y_{2}^{−17/5}, y_{1}^{4}y_{2}^{−12/5} + 4y_{1}^{3}y_{2}^{−12/5}, y_{1}^{5}y_{2}^{−16/5} + y_{1}^{4}y_{2}^{−16/5},
5y_{1}^{6}y_{2}^{−4} + 2y_{1}^{5}y_{2}^{−3} + 2y_{1}^{5}y_{2}^{−4} + 40y_{1}^{4}y_{2}^{−3}.

We are now ready to compute rank_{p}(Horn (B[i, j], c[i, j])).

Figure 2. The supports of the 15 Puiseux polynomial solutions to Horn (B, 0) in Example 6.3

Lemma 6.4. The dimension of the space of Puiseux polynomial solutions of the hypergeometric system Horn (B[i, j], c[i, j]) equals ν_{ij} if the vectors b_{i} and b_{j} are linearly independent and lie in opposite open quadrants of ℤ^{2}.

Proof. Suppose that b_{i} and b_{j} are linearly independent and lie in opposite open quadrants of ℤ^{2}. As in Lemma 6.2, 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 5.2, it is sufficient to compute the number of Puiseux polynomial solutions of H_{B[i,j]}(c[i, j]).

Introduce vectors α, β as follows:

α = \{ \begin{matrix}(b_{i1}, b_{j1}), & if |b_{i1}b_{j2}| > |b_{i2}b_{j1}|, \\ (−b_{i1}, −b_{j1}), & if |b_{i1}b_{j2}| < |b_{i2}b_{j1}|, \end{matrix} β = \{ \begin{matrix}(−b_{i2}, −b_{j2}), & if |b_{i1}b_{j2}| > |b_{i2}b_{j1}|, \\ (b_{i2}, b_{j2}), & if |b_{i1}b_{j2}| < |b_{i2}b_{j1}|. \end{matrix}

Furthermore, denote by R the set of points

R = \{ \begin{matrix}\{(u, v) ∈ ℕ^{2} : u < b_{i2}, v < −b_{j1}\}, & if |b_{i1}b_{j2}| > |b_{i2}b_{j1}|, \\ \{(u, v) ∈ ℕ^{2} : u < b_{i1}, v < −b_{j2}\}, & if |b_{i1}b_{j2}| < |b_{i2}b_{j1}|, \end{matrix}

and call it the base rectangle of H_{B[i,j]}(c[i, j]). By a path connecting two points a, ã ∈ ℕ^{2} we mean a sequence a_{1}, …, a_{k} ∈ ℕ^{2} such that a_{1} = a, a_{k} = ã and the difference a_{i+1} − a_{i} is one of the vectors α, −α, β, −β. We say that a path is increasing if the differences are always one of α, β, and that the path is decreasing the differences are always one of −α, −β. 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_{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_{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_{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_{B[i,j]}(c[i, j]) equals the number of lattice points in R, that is ν_{ij} = min(|b_{i1}b_{j2}|, |b_{i2}b_{j1}|). □

Combining Lemmas 6.1 and 6.4, we obtain a formula for the dimension of the space of Puiseux polynomial solutions of Horn (B, c).

Theorem 6.5. For a generic parameter c,

rank_{p}(Horn (B, c)) = ∑ ν_{ij} ,

where the sum runs over pairs of rows b_{i} and b_{j} of B that are linearly independent and lie in opposite open quadrants of ℤ^{2}.

Notice that Corollary 5.2, Theorem 3.5.1 in [26], Theorem 6.5, and Theorem 5.3 imply the following result:

Theorem 6.6. If c is generic, then

rank (H_{B}(c)) = rank (Horn (B, c)) ≥ g · vol (A) + ∑ ν_{ij} ,

where the sum runs over linearly independent rows of B lying in opposite open quadrants.

7. Regularity and holonomic rank of hypergeometric systems arising from lattices

In this section we prove that, for a generic parameter vector β = A · c, the hypergeometric system I_{B} + ⟨A · θ − β⟩ is regular holonomic, without restriction on the number of columns of the matrix B. Observe that we do need our assumption that the columns of B add up to zero. Notice that genericity of c implies genericity of β.

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_{B} and ⟨A · θ − β⟩, already has dimension n. In order to prove regularity, we need to look at the hypergeometric systems arising from the primary components of I_{B}.

Let ρ be a partial character of L/L_{B}, and let I_{ρ} be as in Section 4. Define H_{ρ}(A · c) = I_{ρ} + ⟨A · θ − A · β⟩. In particular, since ρ_{0} is the trivial character, H_{ρ_{0}}(A · c) = H_{A}(A · c).

Lemma 7.1. For ρ, ρ′ ∈ G_{B}, the group of partial characters of L/L_{B}, the D-modules H_{ρ}(β) and H_{ρ′}(β) are isomorphic.

Proof. It is enough to consider the case when ρ′ = ρ_{0}, so that I_{ρ′} = I_{ρ_{0}} = I_{A}. Given any partial character ρ : L → ℂ^{∗}, let p_{ρ} be any point in X_{ρ} all of whose coordinates are nonzero. We define the map τ_{ρ} : D → D by setting

τ_{ρ}(∑ x^{α}∂^{β}) = ∑ p_{ρ}^{α−β}x^{α}∂^{β}.

It is straightforward to check that τ_{ρ} defines an endomorphism of D, which is clearly an isomorphism. It is also easily checked that τ_{ρ}(I_{A}) = I_{ρ}, and τ_{ρ}(⟨A · θ − β⟩) = ⟨A · θ − β⟩, so that τ_{ρ}(H_{A}(β)) = H_{ρ}(β) and the D-modules D/H_{A}(β) and D/H_{ρ}(β) are isomorphic. □

Corollary 7.2. If ρ ∈ G_{B}, the D-module D/H_{ρ}(A · c) is regular holonomic for all c ∈ ℂ^{n}.

Proof. Hotta has shown (see [19]) that D/H_{A}(A · c) is regular holonomic for all parameters c ∈ ℂ^{n}, since the condition that the sum of the rows of B equals zero implies that the vector (1, 1, …, 1) ∈ ℤ^{n} belongs to the row-span of A. Now apply Lemma 7.1. □

We have shown that the hypergeometric systems arising from the primary components of the lattice ideal I_{B} are regular holonomic for all parameters. This implies that the solutions of these systems belong to the Nilsson class [2, Ch. 6.4]. We will show that the solutions of the hypergeometric system I_{B} + ⟨A · θ − β⟩ satisfy the same properties.

Recall that I_{B} = ∩_{ρ∈G_{B}}I_{ρ}, where G_{B} is the order g group of partial characters, with corresponding ideals I_{ρ}. For any J ⊆ G_{B}, we denote by I_{J} the intersection ∩_{ρ∈J} I_{ρ}. We first need the following result.

Proposition 7.3. Let w ∈ ℕ^{n}\{0}. For generic β, the map

D/(I_{J} + ⟨A · θ − β − A · w⟩) ⟶^{· ∂^{w}} D/(I_{J} + ⟨A · θ − β⟩) ,

given by right multiplication by ∂^{w}, is an isomorphism of left D-modules.

Proof. It is sufficient to consider the case when w = e_{i}, so that our map is right multiplication by ∂_{i}. In order to use the exact argument of the proof of [26, Theorem 4.5.10] (the analogous result for A-hypergeometric systems), we need to show that there exists a nonzero parametric b-function (see [26, Section 4.4]), that is, we need to prove that the following elimination ideal in the polynomial ring ℂ[s_{1}, …, s_{d}] = ℂ[s]

(D[s] I_{J} + ⟨A · θ − s⟩ + D[s] ⟨∂_{i}⟩) ∩ ℂ[s]

is nonzero, where D[s] is the parametric Weyl algebra. In order to do this, we first go through an intermediate step:

(D[s] I_{J} + ⟨A · θ − s⟩ + D[s] ⟨∂_{i}⟩) ∩ ℂ[θ, s] ⊇ (D[s](I_{B} + ⟨∂_{i}⟩) + ⟨A · θ − s⟩) ∩ ℂ[θ, s]
= (D[s](in_{−e_{i}}(I_{B} + ⟨∂_{i}⟩) + ⟨A · θ − s⟩) ∩ ℂ[θ, s]
= in_{(−e_{i},e_{i},0)} (I_{B} + ⟨A · θ − s⟩) ∩ ℂ[θ, s]
⊇ ⟨[θ]_{u} : ∂^{u} ∈ in_{−e_{i}}(I_{B})⟩ + ⟨θ_{i}⟩ + ⟨A · θ − s⟩
⊇ ⟨[θ]_{g u} : ∂^{u} ∈ in_{−e_{i}}(I_{A})⟩ + ⟨θ_{i}⟩ + ⟨A · θ − s⟩ .

Here [θ]_{u} = ∏_{k=1}^{n} ∏_{l=0}^{u_{k}−1}(θ_{k} − l). The first containment holds because I_{B} ⊆ I_{J}. The next equality is true since

I_{B} + ⟨∂_{i}⟩ = in_{−e_{i}}(I_{B}) + ⟨∂_{i}⟩.

The equality in the third line holds by the proof of [26, Theorem 3.1.3], which applies here since I_{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 ∂^{u}, x^{u}∂^{u} = [θ]_{u}. The last containment follows from the fact that g u ∈ L_{B} for all u ∈ ker_{ℤ}(A). Now if we prove that

(⟨[θ]_{g u} : ∂^{u} ∈ in_{−e_{i}}(I_{A})⟩ + ⟨θ_{i}⟩ + ⟨A · θ − s⟩) ∩ 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 ⟨[θ]_{g u} : ∂^{u} ∈ in_{−e_{i}}(I_{A})⟩ + ⟨θ_{i}⟩ + ⟨A · θ − s⟩ onto the s-variables is not surjective.

Observe that the projection of ⟨[θ]_{u} : ∂^{u} ∈ in_{−e_{i}}(I_{A})⟩+⟨θ_{i}⟩+⟨A·θ −s⟩ onto the s-variables is not surjective (by [26, Corollary 4.5.9]). 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. □

Theorem 7.4. For generic β, any solution f of I_{J} + ⟨A · θ − β⟩ can be written as a linear combination

f = ∑_{ρ∈J} f_{ρ} ,

where f_{ρ} is a solution of I_{ρ} + ⟨A · θ − β⟩. In particular, the solutions of I_{B} + ⟨A · θ − β⟩ are linear combinations of the solutions of the systems I_{ρ} + ⟨A · θ − β⟩, for ρ ∈ G_{B}.

Proof. We proceed by induction on the cardinality of J, the base case being trivial. Assume that our conclusion is valid for subsets of G_{B} of cardinality r−1 ≥ 1, pick J ⊆ G_{B} of cardinality r and fix ρ ∈ J.

Let P be an element of I_{J \{ρ}} such that P ∉ I_{ρ}. Since all of the ideals I_{τ}, τ ∈ G_{B}, are homogeneous with respect to the multi-grading given by A, we may assume that P is homogeneous, and write

P = λ_{1}∂^{u^{(1)}} + ⋯ + λ_{p−1}∂^{u^{(p−1)}} + ∂^{w},

where λ_{1}, …, λ_{p−1} ∈ ℂ and A·u^{(1)} = A·u^{(2)} ⋯ = A·u^{(p−1)} = A·w. Notice that the polynomial

P̄ = λ_{1}∂^{u^{(1)}} + ⋯ + λ_{p−1}∂^{u^{(p−1)}} − [λ_{1}ρ(u^{(1)} − w) + ⋯ + λ_{p−1}ρ(u^{(p−1)} − w)]∂^{w}

is an element of the ideal I_{ρ}, since this ideal is generated by all binomials of the form ∂^{α} − ρ(α − γ)∂^{γ}, where A · α = A · γ. To simplify the notation, set −λ to be the coefficient of ∂^{w} in P̄, that is,

λ = λ_{1}ρ(u^{(1)} − w) + ⋯ + λ_{p−1}ρ(u^{(p−1)} − w).

Now let f be a solution of I_{J} +⟨A·θ −β⟩, and consider the function P̄f. For any Q ∈ I_{J \{ρ}}, we have QP̄ ∈ I_{J}. This implies that QP̄f = 0. Furthermore, noting that P̄ is A-homogeneous of multi-degree A · w, we conclude that P̄f is a solution of I_{J \{ρ}} + ⟨A · θ − β − A · w⟩. Since β is generic, so is β + A · w, and by the inductive hypothesis we can write P̄f = ∑_{τ∈J \{ρ}} g_{τ}, where each g_{τ} is a solution of I_{τ} + ⟨A · θ − β − A · w⟩.

By Proposition 7.3, ∂^{w} induces an isomorphism between the solution spaces of I_{τ} +⟨A·θ −β⟩ and I_{τ} + ⟨A · θ − β − A · w⟩, so that we can find a solution g̃_{τ} of I_{τ} + ⟨A · θ − β⟩ such that ∂^{w}g̃_{τ} = g_{τ}. Now

P̄g̃_{τ} = ∑_{i=1}^{p−1} λ_{i}∂^{u(i)}g̃_{τ} − λ∂^{w}g̃_{τ}

= (∑_{i=1}^{p−1} λ_{i}τ(u^{(i)} − w) − λ) g_{τ}.

The last equality holds because g̃_{τ} is a solution of I_{τ}, and therefore ∂^{u^{(i)}} − τ(u^{(i)} − w)∂^{w} annihilates it, yielding ∂^{u^{(i)}} g̃_{τ} = τ(u(i) − w)∂^{w}g̃_{τ} = τ(u^{(i)} − w)g_{τ}.

Notice that the coefficient ∑_{i=1}^{p−1} λ_{i}τ(u^{(i)} − w) − λ is nonzero, for otherwise we could rewrite P̄ using the sum instead of λ, and conclude that P̄ ∈ I_{τ}. But we know P ∈ I_{τ}, so P̄ − P ∈ I_{τ}, a contradiction since this is a nonzero multiple of ∂^{w}, and the ideal I_{τ} contains no monomials. (The fact that P̄ − P ≠ 0 follows from P̄ ∈ I_{ρ} and P ∉ I_{ρ}).

Finally define f_{τ} = (∑_{i=1}^{p−1} λ_{i}τ(u^{(i)} − w) − λ)^{−1} g̃_{τ}, so that f_{τ} is a solution of I_{τ} + ⟨A · θ − β⟩ and

P̄ ∑_{τ∈J \{ρ}} f_{τ} = ∑_{τ∈J \{ρ}} g_{τ} = P̄f.

If h = f − ∑_{τ∈J \{ρ}} f_{τ}, then h is a solution of I_{J} + ⟨A · θ − β⟩ that satisfies P̄h = 0. Now consider P h. Since P ∈ I_{J \{ρ}}, P h is a solution of I_{ρ} +⟨A·θ−β −A·w⟩, and a similar argument as before yields a solution f_{ρ} of I_{ρ} + ⟨A · θ − β⟩ such that P h = P f_{ρ}. Let h̃ = h − f_{ρ}, so that f = ∑ f_{τ} + f_{ρ} + h̃ and P h̃ = 0. But P̄h̃ = P̄h − P̄f_{ρ} = 0 since P̄ ∈ I_{ρ}.

Now P h̃ = P̄h̃ = 0 implies (P − P̄)h̃ = 0, so that ∂^{w}h̃ = 0, because P − P̄ is a nonzero multiple of ∂^{w}. But then h̃ is a solution of I_{J} + ⟨A · θ − β⟩ that is mapped under ∂^{w} to the zero element in the solution space of I_{J} + ⟨A · θ − β − A · w⟩, which, using the genericity of β and Proposition 7.3, implies that h̃ = 0. Thus we have obtained an expression for f as a linear combination of solutions of the systems I_{τ} + ⟨A · θ − β⟩, τ ∈ J, and the proof of the inductive step is finished. □

Considering J = G_{B}, we deduce that all solutions of D/(I_{B} + ⟨A · θ − β⟩) split as a sum of solutions for each I_{ρ}, yielding a kind of converse to Theorem 5.3. We remark that this result is not true without the genericity assumption on β, since for certain parameters (for instance for β = 0, where the constant function 1 is a solution), the solutions to the different ideals H_{ρ}(β) are not linearly independent. Observing that D/(I_{B} + ⟨A · θ − β⟩) is holonomic, we deduce:

Theorem 7.5. Suppose that B has zero column sums. The D-module associated with the lattice L_{B} and a parameter β ∈ ℂ^{d}, D/(I_{B} + ⟨A · θ − β⟩), is regular holonomic for generic choices of the parameter β.

8. Holonomicity and regularity of the Horn system H_{B}(c)

In this section assume that m = 2. Our goal is to investigate for which parameters c the D-module D/H_{B}(c) is regular holonomic. Our major tools will be the explicit description of the primary decomposition of I given in Section 4, the regular holonomicity of the hypergeometric systems associated to lattices proved in the previous section, and the fact that regular holonomic D-modules form an abelian category (see [3]).

First let us show that H_{B}(c) is holonomic for generic c.

Theorem 8.1. Let m = 2 and c generic parameter vector. Then H_{B}(c) is holonomic.

Proof. Write I = ⟨∂^{u_{+}} − ∂^{u_{−}}, ∂^{v_{+}} − ∂^{v_{−}}⟩, where u and v are the columns of B. Consider first the case when B has no linearly dependent rows in opposite open quadrants of ℤ^{2}. Then the ring

\frac{ℂ[x_{1}, …, x_{n}, z_{1}, …, z_{n}]}{⟨z^{u_{+}} − z^{u_{−}}, z^{v_{+}} − z^{v_{−}}⟩ + ⟨∑_{j=1}^{n} a_{ij}x_{j}z_{j} : i = 1, …, n − m⟩}

has dimension n (see Lemma 11.1). Since the polynomial ring modulo the characteristic ideal of H_{B}(c) is a subring of this one, we conclude that H_{B}(c) is holonomic for all c ∈ ℂ^{m}.

Now assume that B has linearly dependent rows b_{i}, b_{j} in opposite open quadrants of ℤ^{2}. In this case, the ideal ⟨z^{u_{+}} − z^{u_{−}}, z^{v_{+}} − z^{v_{−}}⟩ + ⟨∑ a_{ij}x_{j}z_{j} : j = 1, …, n − m⟩ will have a lowerdimensional component corresponding to the vanishing of z_{i} and z_{j}, by the results in Section 4 about primary decomposition of codimension 2 lattice basis ideals.

To ensure holonomicity of H_{B}(c), we will construct, for each pair b_{i}, b_{j} of linearly dependent rows of B in opposite open quadrants of ℤ^{2}, an element of the ideal H_{B}(c) that contains no x_{i}, x_{j}, ∂_{i}, ∂_{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 ∈ ℚ, r ∈ ℂ, such that p θ_{1} + q θ_{2} − r lies in H_{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 ∈ L_{B} such that w_{1} = w_{2} = 0. Then we can find two monomials m_{1}, m_{2} in ℂ[∂] with disjoint supports, that are not divisible by either ∂_{1} or ∂_{2} such that ∂_{1}^{k}m_{1}(∂^{w_{+}} − ∂^{w_{−}}) ∈ I for some k > 0 and ∂_{2}^{l}m_{2}(∂^{w_{+}} − ∂^{w_{−}}) ∈ I for some l > 0. This follows from the arguments that proved Proposition 4.4. Call μ = m_{1}(∂^{w_{+}} − ∂^{w_{−}}) and λ = m_{2}(∂^{w_{+}} − ∂^{w_{−}}). Notice that μ, λ do not depend on ∂_{1}, ∂_{2}. Then, using x_{1}^{k}∂_{1}^{k} = θ_{1}(θ_{1} − 1) ⋯ (θ_{1} − k + 1) = [θ_{1}]_{k} we see that [θ_{1}]_{k}μ ∈ H_{B}(c). Similarly, [θ_{2}]_{l}λ ∈ H_{B}(c).

Consider the left ideal in the Weyl algebra generated by:

p θ_{1} + q θ_{2} − r, [θ_{1}]_{k}μ, [θ_{2}]_{l}λ.

This ideal is contained in H_{B}(c). Now notice that θ_{1}, θ_{2}, λ and μ are pairwise commuting elements of D_{n}. This means that we can think of ⟨p θ_{1} + q θ_{2} − r, [θ_{1}]_{k}μ, [θ_{2}]_{l}λ⟩ as an ideal in ℂ[θ_{1}, θ_{2}, ∂_{3}, …, ∂_{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 θ_{1}, θ_{2}, ∂_{3}, …, ∂_{n}.

Finding the element of H_{B}(c) that we want has now been reduced to eliminating θ_{1} and θ_{2} from

(14)  ⟨p θ_{1} + q θ_{2} − r, [θ_{1}]_{k}μ, [θ_{2}]_{l}λ⟩ ⊂ ℂ[θ_{1}, θ_{2}, ∂_{3}, …, ∂_{n}, r].

Since the geometric counterpart of elimination is projection, in order to check that the elimination ideal

⟨p θ_{1} + q θ_{2} − r, [θ_{1}]_{k}μ, [θ_{2}]_{l}λ⟩ ∩ ℂ[λ, μ, r]

is nonzero, we need to show that there exist complex numbers ∂_{3}, …, ∂_{n} and r such that, for all values of θ_{1}, θ_{2} ∈ ℂ, the tuple (θ_{1}, θ_{2}, ∂_{3}, …, ∂_{n}, r) is not a solution of (14). If (∂_{3}, …, ∂_{n}) is generic, the polynomials μ and λ evaluated at that point will be nonzero. Thus, in order for [θ_{1}]_{k}μ to vanish, θ_{1} must be an integer between 0 and k. Analogously, θ_{2} must be an integer between 0 and l. But then, for most values of r, p θ_{1} + q θ_{2} − r is nonzero. Thus, the projection of the zero set of (14) onto the ∂_{3}, …, ∂_{n}, r coordinates is not surjective. This implies that (14) contains an element P that does not depend on θ_{1} or θ_{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_{B}(c), that does not depend on x_{1}, x_{2}, ∂_{1}, ∂_{2}, and is nonzero for generic c. □

Example 8.2. Consider the matrix

B = ( \begin{matrix}1 & 2 \\ −2 & −4 \\ 1 & 1 \\ 0 & 1 \end{matrix} ) .

To prove that H_{B}(c) is holonomic for generic c, we need to find an element of H_{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_{B} with its first two coordinates equal to zero. The vector (0, 0, −1, 1) works. It is easy to check that ∂_{1}^{2}∂_{3}^{2}(∂_{3} − ∂_{4}) and ∂_{2}^{4}(∂_{3} − ∂_{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 θ_{1} and θ_{2} from:

⟨θ_{1}(θ_{1} − 1)∂_{3}^{2}(∂_{3} − ∂_{4}), θ_{2}(θ_{2} − 1)(θ_{2} − 2)(θ_{2} − 3)(∂_{3} − ∂_{4}), 2θ_{1} + θ_{2} − r⟩ ⊆ ℂ[θ_{1}, θ_{2}, ∂_{3}, ∂_{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)∂_{3}^{2}(∂_{3} − ∂_{4}) ∈ H_{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.

Now we need to tackle the question of regularity. Our strategy is to reduce the problem of regularity of H_{B}(c) to proving regularity of related systems. The first step is the following result.

Lemma 8.3. Let α be as in Proposition 4.4. For generic c, the sequence (15)

0 → D/(I_{B} + ⟨A · θ − A · (c + α)⟩) ⟶^{· ∂^{α}} D/H_{B}(c) ⟶^{π} D/(I + ⟨∂^{α}⟩ + ⟨A · θ − A · c⟩) → 0,

where π is the natural projection, is exact.

Proof. The only part of exactness that is not clear is that right multiplication by ∂^{α} is injective (it is well defined since ∂^{α}I_{B} ⊆ I). To see this, consider the following commutative diagram:

\begin{matrix}0 & \overline{} & D/(I_{B} + ⟨A · θ − A · (c + α)⟩) & \overline{· ∂^{α}_{}} & D/(I_{B} + ⟨A · θ − A · c⟩) & ^{}0 \\ D/H_{B}(c) & & ^{· ∂^{α} ^{}} \end{matrix}

where the vertical arrow is the natural inclusion. The upper row of the diagram is exact by Theorem 7.3, since c is generic. But then the commutativity implies that the diagonal arrow is injective. □

Since the category of regular holonomic D-modules is an abelian category [3, pp. 303,308,316], the previous lemma gives us an indication of how to prove that D/H_{B}(c) is a regular holonomic D-module, namely, we have to show that both D/(I_{B} + ⟨A · θ − A · (c + α)⟩) and D/(I + ⟨∂^{α}⟩ + ⟨A · θ − A · (c + α)⟩) are regular holonomic. For generic c, we already know that this is true for the first module (Theorem 7.5). We will now prove that the second module is regular holonomic (Corollary 8.7). We start with an auxiliary result.

Lemma 8.4. Let u, v ∈ ℕ^{n} such that ⟨∂^{u}, ∂^{v}⟩ is a complete intersection. If c is generic, then

⟨∂^{u}, ∂^{v}⟩ + ⟨A · θ − A · c⟩

is a regular holonomic system of differential equations, whose solution space has a basis of Puiseux monomials.

Proof. It is enough to show that the system

⟨x^{u}∂^{u}, x^{v}∂^{v}⟩ + ⟨A · θ − A · c⟩

satisfies the desired properties since x^{u} and x^{v} are units in ℂ(x).

Now

⟨x^{u}∂^{u}, x^{v}∂^{v}⟩ + ⟨A · θ − A · c⟩ = ⟨[θ]_{u}, [θ]_{v}⟩ + ⟨A · θ − A · c⟩ = D · F,

where

[θ]_{u} = ∏_{k=1}^{n} ∏_{l=0}^{u_{k}−1}(θ_{k} − l) ,

and

F = ⟨[θ]_{u}, [θ]_{v}⟩ + ⟨A · θ − A · c⟩ ⊆ ℂ[θ].

This means that D · F is a Frobenius ideal (see [26, Section 2.3]). By [26, Proposition 2.3.6, Theorem 2.3.11], if we can show that F is artinian and radical, it will follow that D · F is holonomic, with solution space spanned by {x^{p} : p ∈ V(F)}, where V(F) is the zero set of the ideal F ⊆ ℂ[θ], and we will be done.

To show that F is artinian and radical, we proceed as in [26, Theorem 3.2.10]. Let p ∈ V(F). Then there exist 1 ≤ i < j ≤ 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 [θ]_{u}(p) = [θ]_{v}(p) = 0 and the fact that u and v have disjoint supports, because ⟨∂^{u}, ∂^{v}⟩ is a complete intersection. Since c is generic, the minor of A complementary to {i, j} must be nonzero (otherwise the equations θ_{i} = p_{i}, θ_{j} = p_{j} and A · θ = A · c would be incompatible). Hence its i-th and j-th coordinates determine p uniquely in V(F). □

Remark 8.5. If all maximal minors of A are nonzero, the above lemma holds without restriction on c.

Theorem 8.6. Write I = ⟨∂^{u_{+}} − ∂^{u_{−}}, ∂^{v_{+}} − ∂^{v_{−}}⟩, where u and v are the columns of B. Let ∂^{α} be a monomial satisfying:

(16)  α_{i} > 0 ⟹ u_{i} > 0.

Then, for generic c, the D-ideal I + ⟨∂^{α}⟩ + ⟨A · θ − A · c⟩ has only Puiseux polynomial solutions.

Proof. We proceed by induction on |α| = α_{1}+⋯+α_{n}, the length of α. If |α| ≤ min{u_{i} : u_{i} > 0}, in particular, if |α| = 1 (recall that |u| = 0), then ∂^{α} divides ∂^{u_{+}}, so that all solutions of I + ⟨∂^{α}⟩ + ⟨A · θ − A · c⟩ are solutions of ⟨∂^{α}, ∂^{u_{−}}⟩ + ⟨A · θ − A · c⟩. But the latter ideal has only Puiseux polynomial solutions by Lemma 8.4, since c is generic.

Assume now that our result is true for length s and let α be of length s + 1 satisfying (16). Choose i such that α_{i} > 0 (and so u_{i} > 0), and let φ be a solution of I + ⟨∂^{α}⟩ + ⟨A · θ − A · c⟩. The function ∂_{i}φ is a solution of I + ⟨∂^{α−e_{i}}⟩ + ⟨A · θ − A · c − A · e_{i}⟩. But |α − e_{i}| = s and c + e_{i} is still generic, so the inductive hypothesis implies that ∂_{i}φ is a Puiseux polynomial. Write:

∂_{i}φ = ∑_{l=0}^{N_{0}} g_{l}^{(0)}x_{i}^{l} + ∑_{l=0}^{N_{1}} g_{l}^{(1)}x_{i}^{μ_{1}+l} + ⋯ + ∑_{l=0}^{N_{t}} g_{l}^{(t)}x_{i}^{μ_{t}+l},

where the g_{l}^{(k)} are Puiseux polynomials, constant with respect to x_{i}, t is a natural number, and μ_{1}, …, μ_{t} ∈ ℂ are nonintegers with noninteger pairwise differences. Then

(17) φ = ∑_{l=0}^{N_{0}} g_{l}^{(0)} \frac{x_{i}^{l+1}}{l + 1} + ∑_{l=0}^{N_{1}} g_{l}^{(1)} \frac{x_{i}^{μ_{1}+l+1}}{μ_{1} + l + 1} + ⋯ + ∑_{l=0}^{N_{t}} g_{l}^{(t)} \frac{x_{i}^{μ_{t}+l+1}}{μ_{t} + l + 1} + G(x_{1}, …, x̂_{i}, …, x_{n}).

If we prove that G is a Puiseux polynomial, it will follow that so is φ, and the proof will be finished. We know that φ is a solution of ⟨A · θ − A · c⟩. By construction, so is φ − G. Then G is a solution of ⟨A · θ − A · c⟩. Recall that ∂_{i}G = 0.

We also know that ∂^{u_{+}}φ = ∂^{u_{−}}φ. We want to compare the coefficients of the integer powers of x_{i} in the expressions we obtain by applying ∂^{u_{+}} and ∂^{u_{−}} to (17). Since we are only looking at the integer powers of x_{i}, we need only look at ∑_{l=0}^{N_{0}} g_{l}^{(0)}(x_{i}^{l+1}/(l + 1)) + G.

(18) ∂^{u_{+}} (∑_{l=0}^{N_{0}} g_{l}^{(0)} \frac{x_{i}^{l+1}}{l + 1} + G) = ∑_{l=0}^{N_{0}} \frac{(l + 1)l(l − 1) ⋯ (l + 2 − u_{i})}{l + 1}(∂^{u_{+}−u_{i}e_{i}}g_{l}^{(0)})x_{i}^{l+1−u_{i}}.

Notice that there is no G in the above expression, since ∂_{i}G = 0 and u_{i} > 0. Also, the highest power of x_{i} appearing in (18) is x_{i}^{N_{0}+1−u_{i}}.

(19)  ∂^{u_{−}} (∑_{l=0}^{N_{0}} g_{l}^{(0)} \frac{x_{i}^{l+1}}{l + 1} + G) = ∑_{l=0}^{N_{0}}(∂^{u_{−}}g_{l}^{(0)})\frac{x_{i}^{l+1}}{l + 1} + ∂^{u_{−}}G.

We equate the coefficients of x_{i}^{l+1} in (18) and (19) to obtain:

(20)  \frac{∂^{u_{−}}g_{l}^{(0)}}{l + 1} = (l + u_{i}) ⋯ (l + 2)∂^{u_{+}−u_{i}e_{i}}g_{l+u_{i}}^{(0)} , for l = 0, …, N_{0} − u_{i}.

If l = N_{0} + 1 − u_{i}, …, N_{0}, then ∂^{u_{−}}g_{l}^{(0)} = 0. Also,

∂^{u_{−}}G = (u_{i} − 1)(u_{i} − 2) ⋯ 2 · 1 · ∂^{u_{+}−u_{i}e_{i}}g_{u_{i}−1}^{(0)}.

Applying ∂^{u_{−}} to (20), we see that

∂^{2u_{−}}g_{l}^{(0)} = (l + u_{1}) ⋯ (l + 2)(l + 1)∂^{u_{+}−u_{i}e_{i}}∂^{u_{−}}g_{l+u_{i}}^{(0)} = 0 , for l = N_{0} + 1 − 2u_{i}, …, N_{0} − u_{i}.

Applying ∂^{u_{−}} enough times, we conclude that, if ku_{i} > N_{0} + 1, then ∂^{ku_{−}}G = 0. But now, G is a solution of ⟨∂_{i}, ∂^{ku_{−}}⟩ + ⟨A · θ − A · c⟩, and c is generic. By Lemma 8.4, G is a Puiseux polynomial. □

Corollary 8.7. For generic c and α satisfying (16), in particular, if α is as in Proposition 4.4, the D-ideal I + ⟨∂^{α}⟩ + ⟨A · θ − A · c⟩ is regular holonomic.

Proof. Holonomicity follows since H_{B}(c) is holonomic, and regularity is a consequence of the previous theorem. □

We can finally prove the main result in this section.

Theorem 8.8. Assume that all column sums of B are zero. For generic c, the Horn system H_{B}(c) is regular holonomic.

Proof. Follows from Lemma 8.3, Theorem 7.5 and Corollary 8.7 using the fact that regular holonomic D-modules form an abelian category. □

9. Indicial ideals and holonomic ranks

Now that we know that the Horn system H_{B}(c) and the hypergeometric system I_{B}+⟨A·θ−A·c⟩ 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 [26, Chapters 1,2 and 3]. 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_{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 · in_{(−w,w)}(J) ∩ ℂ[θ_{1}, …, θ_{n}]

is a a zero dimensional ideal of ℂ[θ_{1}, …, θ_{n}], whose zeros are the so-called 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 [26, Theorem 2.3.9, Theorem 2.5.1].

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 ([26, Corollary 3.1.6]), and the same ideas work for the case of Horn systems and hypergeometric systems arising from lattices.

Theorem 9.1. For generic parameters c, we have

ind_{w}(H_{B}(c)) = ((R · in_{w}(I)) ∩ ℂ[θ]) + ⟨A · θ − A · c⟩,

and

ind_{w}(I_{B} + ⟨A · θ − A · c⟩) = ((R · in_{w}(I_{B})) ∩ ℂ[θ]) + ⟨A · θ − A · c⟩,

Proof. The proof of the analogous fact for A-hypergeometric systems follows from [26, Theorem 3.1.3 and Proposition 3.1.5]. But [26, Proposition 3.1.5] carries over to the cases that interest us without any modification in its proof. Moreover the proof of [26, Theorem 3.1.3] 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_{B} satisfy. □

Our next goal is to compute the primary decomposition of the indicial ideals of H_{B}(c) and I_{B} + ⟨A · θ − A · c⟩ 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.

Definition 9.2. Let M be a monomial ideal in the polynomial ring ℂ[∂_{1}, …, ∂_{n}]. A standard pair of M is a pair (∂^{η}, σ), where σ is a possibly empty subset of {1, …, n}, that satisfies

i. η_{i} = 0 for all i ∈ σ; ii. for any choice of integers μ_{j} ≥ 0, j ∈ σ, the monomial ∂^{η} ∏_{j∈σ} ∂_{j}^{μ_{j}} is not in M; iii. for all l ∉ σ, there exist integers μ_{l} ≥ 0 and μ_{j} ≥ 0, j ∈ σ, such that ∂^{η}∂_{l}^{μ_{l}} ∏_{j∈σ} ∂_{j}^{μ_{j}} lies

in M.

We denote the set of standard pairs of a monomial ideal M by S(M). By [29, Equation (3.2)],

M = ⋂_{(∂^{η},σ)∈S(M)}⟨∂_{i}^{η_{i}+1} : i ∉ σ⟩.

The prime ideal ⟨∂_{i} : i ∉ σ⟩ is associated to M if and only if there exists a standard pair of the form (·, σ) in S(M). A standard pair (∂^{η}, σ) is called top dimensional if ⟨∂_{i} : i ∉ σ⟩ is a minimal associated prime of M, it is called 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_{B} are unmixed (I is a complete intersection, and the associated primes of I_{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 [20, Corollary 1]), and the same holds for I_{B}. This means that a standard pair (∂^{η}, σ) of either in_{w}(I) or in_{w}(I_{B}) is top dimensional if and only if #σ = d.

Let T (in_{w}(I)) be the set of top dimensional standard pairs (∂^{η}, σ) of in_{w}(I) such that the rows of B indexed by i ∉ σ are linearly independent.

Notice that if (∂^{η}, σ) is a top-dimensional standard pair of in_{w}(I_{B}), then the rows of B indexed by i ∉ σ are linearly independent (the proof of [22, Lemma 2.3] works for lattice ideals too). Then T (in_{w}(I_{B})) equals the set of top-dimensional standard pairs of in_{w}(I_{B}).

Given a standard pair in either T (in_{w}(I)) or T (in_{w}(I_{B})), and an arbitrary parameter vector c, there exists a unique vector v such that A · v = A · c, and v_{k} = η_{k}, v_{l} = η_{l}.

Suppose that (∂^{η}, σ) is a standard pair of in_{w}(I) that does not belong to T (in_{w}(I)). Then either #σ < m or #σ = n − 2 and the columns of B corresponding to the indices not in σ are linearly dependent. In both of these cases, for a generic choice of c, the system A · v = A · c, v_{i} = η_{i} for i ∉ σ, has no solutions. The same holds for standard pairs not in T (in_{w}(I_{B})).

We can now describe the primary decomposition of the indicial ideals of H_{B}(c) and I_{B} + ⟨A · θ − A · c⟩ with respect to w, in analogy to [26, Theorem 3.2.10].

Proposition 9.3. For a generic parameter c, the indicial ideal of H_{B}(c) with respect to w equals the following intersection of maximal ideals:

(21)  ⋂_{(∂^{η},σ)∈T (in_{w}(I))} (⟨θ_{i} − η_{i} : i ∉ σ⟩ + ⟨A · θ − A · c⟩),

and the indicial ideal of I_{B} + ⟨A · θ − A · c⟩ equals:

(22)  ⋂_{(∂^{η},σ)∈T (in_{w}(I_{B}))} (⟨θ_{i} − η_{i} : i ∉ σ⟩ + ⟨A · θ − A · c⟩).

Proof. We prove the statement for the indicial ideal of H_{B}(c). The other indicial ideal is computed in exactly in the same manner.

By [26, Corollary 3.2.3], the indicial ideal is

J = ⟨A · θ − A · c⟩ + ⋂_{(∂^{η},σ)∈S(in_{w}(I))}⟨θ_{i} − η_{i} : i ∉ σ⟩.

It is clear that the ideal (21) 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·v = A·c, and for some (∂^{η}, σ) ∈ S(in_{w}(I)), we have that v_{i} = η_{i} for all i ∉ σ. Since our parameter c is generic, we must have that (∂^{η}, σ) belongs to T (in_{w}(I)). These are exactly the roots of the ideal (21). It also follows from the genericity of c that all the zeros of J are distinct, and the proof is finished. □

Notice that the degree of in_{w}(I) is d_{1} · 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} · d_{2}. This and the previous proposition imply the following result. Here we use very strongly that H_{B}(c) is regular holonomic, since this property allows us to conclude that the holonomic rank of H_{B}(c) equals the degree of its indicial ideal.

Corollary 9.4. Let ν be the sum of the multiplicities of the minimal primes of in_{w}(I) corresponding to linearly dependent sets of two rows of B. For a generic parameter vector c, the degree of the fake indicial ideal is exactly d_{1} · d_{2} − ν. Therefore,

rank (H_{B}(c)) = rank (Horn (B, c)) = d_{1} · d_{2} − ν = #T (in_{w}(I)).

Our desired formula for the generic rank of a bivariate Horn system now follows from Proposition 4.2.

Theorem 9.5. For generic c and m = 2,

rank (H_{B}(c)) = rank (Horn (B, c)) = d_{1} · d_{2} − ∑ ν_{ij} ,

where the sum runs over linearly dependent rows of B that lie in opposite open quadrants of ℤ^{2}.

Proof. By Proposition 4.2, the sum of the multiplicities of the minimal primes of I corresponding to linearly dependent rows of B is at least the sum of the corresponding indices ∑ ν_{ij}. This implies that

rank (H_{B}(c)) = rank (Horn (B, c)) ≤ d_{1} · d_{2} − ∑ ν_{ij},

where the sum runs over linearly dependent rows of B lying in opposite open quadrants of ℤ^{2}. The reverse inequality follows from Theorem 6.6. □

We now have an explicit description of a basis of the solution space of Horn (B, c) (and H_{B}(c)).

Theorem 9.6. If c is generic, the fully supported series obtained by applying the isomorphism from Corollary 5.2 to the fully supported series constructed in Theorem 5.3 and the Puiseux polynomials constructed in Theorem 6.5 form a basis for the solution space of Horn (B, c).

Proof. Theorem 5.3 and Corollary 5.2 give us g · vol (A) + ∑ ν_{ij} linearly independent solutions of H_{B}(c) (here the sum runs over linearly independent rows of B). By Theorem 9.5, these must span the solution space of H_{B}(c). □

Notice that we can use Theorem 9.5 to conclude that the multiplicities of the minimal primes ⟨∂_{i}, ∂_{j}⟩ of I corresponding to linearly dependent rows of B are preserved under Gröbner deformation.

Corollary 9.7. If b_{i} and b_{j} are linearly dependent rows of B in opposite open quadrants of ℤ^{2}, and w ∈ ℝ^{n} is a generic weight vector, then the multiplicity of ⟨∂_{i}, ∂_{j}⟩ as an associated prime of in_{w}(I) equals the index ν_{ij}.

The exactly same methods that proved Theorem 9.5 will compute the rank of the hypergeometric system arising from a lattice (actually, this proof is easier, since #T (in_{w}(I_{B})) = deg(I_{B}) = g · vol (A) is easier to compute than #T (in_{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_{B}.

Theorem 9.8. For generic c,

rank (I_{B} + ⟨A · θ − A · c⟩) = #T (in_{w}(I_{B})) = deg(I_{B}) = g · vol (A).

We also have an explicit description of the solution space of I_{B} + ⟨A · θ − A · c⟩ for generic c as a consequence of the previous theorem and Theorem 7.4.

Corollary 9.9. For generic c the solution space of I_{B} + ⟨A · θ − A · c⟩ equals the direct sum of the solution spaces of the systems I_{ρ} + ⟨A · θ − A · c⟩, for all ρ in the group G_{B} of partial characters.

10. Regularity and holonomicity of Horn (B, c)

Throughout this section we assume that m = 2. Since we do not have a D-module isomorphism between H_{B}(c) and Horn (B, c), the regular holonomicity of H_{B}(c) does not directly prove that Horn (B, c) is regular holonomic. In this section we prove that the bivariate hypergeometric system Horn (B, c) is regular holonomic.

We first show that the system Horn (B, c) is holonomic for generic c. Recall that a system of differential equations is said to be 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

(23)  \begin{matrix}H_{1} = Q_{1}(θ) − y_{1}P_{1}(θ), \\ H_{2} = Q_{2}(θ) − y_{2}P_{2}(θ). \end{matrix}

By the definition of the Horn system (see Section 2) the bivariate polynomials P_{i}, Q_{i} satisfy the compatibility condition

(24)  R_{1}(s + e_{2})R_{2}(s) = R_{2}(s + e_{1})R_{1}(s),

where R_{i}(s) = P_{i}(s)/Q_{i}(s + e_{i}) and {e_{1}, e_{2}} is the standard basis of ℤ^{2}.

Theorem 10.1. A bivariate Horn system with generic parameters is holonomic.

To prove this theorem we need some intermediate results and notation. Let (H_{1}, H_{2}) ⊂ D_{2} denote the ideal generated by the hypergeometric operators defining the Horn system. By σ(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 (23) 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 2) the greatest common divisor of the principal symbols of H_{1} and H_{2} is given by a product of powers of linear forms ay_{1}z_{1}+by_{2}z_{2}, where a, b ∈ ℤ. Thus to prove Theorem 10.1 it suffices to show that for any a, b ∈ ℤ such that ay_{1}z_{1}+by_{2}z_{2} divides gcd(σ(H_{1}), σ(H_{2})) there exists an operator T_{a,b} ∈ (H_{1}, H_{2}) whose principal symbol is not divisible by ay_{1}z_{1} + by_{2}z_{2}.

Remark 10.2. For generic parameters the compatibility condition (24) is equivalent to the relations

(25)  [y_{1}P_{1}(θ), y_{2}P_{2}(θ)] = 0, (E_{2}Q_{2})(θ)(E_{1}E_{2}Q_{1})(θ) = (E_{1}Q_{1})(θ)(E_{1}E_{2}Q_{2})(θ),

where [ ,] denotes the commutator of two operators, (E_{i}^{λ}P)(s) = P (s+λe_{i}) and E_{i} = E_{i}^{1}. Indeed, the equalities (25) mean that the numerators (respectively the denominators) of the rational functions in (24) are equal. The generic parameters assumption implies that no cancellations can occur and hence this is indeed the case.

Lemma 10.3. For any α, β, γ, δ ∈ ℂ and P_{1}(θ), P_{2}(θ), Q_{1}(θ), Q_{2}(θ) satisfying the relations

(26)  [y_{1}P_{1}(θ), y_{2}P_{2}(θ)] = 0, (E_{2}Q_{2})(θ)(E_{1}E_{2}Q_{1})(θ) = (E_{1}Q_{1})(θ)(E_{1}E_{2}Q_{2})(θ),

it holds that:

(27) (α(E_{2}^{−1}Q_{1})(θ) − βy_{1}P_{1}(θ))(γQ_{2}(θ) − δy_{2}P_{2}(θ))−

(α(E_{1}^{−1}Q_{2})(θ) − βy_{2}P_{2}(θ))(γQ_{1}(θ) − δy_{1}P_{1}(θ)) = | \begin{matrix}α & β \\ γ & δ \end{matrix} | Ψ,

where Ψ = y_{1}Q_{2}(θ)P_{1}(θ) − y_{2}Q_{1}(θ)P_{2}(θ).

The proof of Lemma 10.3 is a direct computation which uses the compatibility conditions (26) and the Weyl algebra identity (E_{i}^{−1}Q_{j})(θ)y_{i} = y_{i}Q_{j}(θ).

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:

(28)  \begin{matrix}U_{1} = f(t)Q_{1}(θ) − y_{1}g(t)P_{1}(θ), \\ U_{2} = f(t)Q_{2}(θ) − y_{2}g(t)P_{2}(θ), \end{matrix}

where f, g are arbitrary non zero univariate polynomials, t = θ_{1} + θ_{2} and P_{i}, Q_{i} are arbitrary bivariate polynomials such that deg f + deg Q_{i} = deg g + deg P_{i} and that P_{i}, Q_{i} satisfy (26). 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}(θ), Q_{i}(θ), 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 (28), i.e., to construct an operator in the ideal (U_{1}, U_{2}) whose principal symbol is not divisible by σ(t) = y_{1}z_{1} + y_{2}z_{2}. We do it as follows.

Lemma 10.4. Let Ψ be as in Lemma 10.3. Then R(f(t), g(t))Ψ ∈ (U_{1}, U_{2}), where R(f(t), g(t)) is the resultant of f, g.

Proof. Let us write the polynomials f, g in the form f(t) = ∑_{i=0}^{d} f_{i}t^{i}, g(t) = ∑_{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 (27), and the fact that the subring of the Weyl algebra generated by θ_{1} and θ_{2} is commutative, we conclude that for any j = 0, …, d

∑_{i=0}^{d} | \begin{matrix}f_{j} & g_{j} \\ f_{i} & g_{i} \end{matrix} | Ψt^{i} = ∑_{i=0}^{d} Θ_{1j}(f_{i}t^{i}Q_{2}(θ) − y_{2}g_{i}t^{i}P_{2}(θ)) − ∑_{i=0}^{d} Θ_{2j}(f_{i}t^{i}Q_{1}(θ) − y_{1}g_{i}t^{i}P_{1}(θ))
= Θ_{1j}U_{2} + Θ_{2j}U_{1} ∈ (U_{1}, U_{2}),

where Θ_{1j} = f_{j}(E_{2}^{−1}Q_{1})(θ) − g_{j}y_{1}P_{1}(θ) and Θ_{2j} = f_{j}(E_{1}^{−1}Q_{2})(θ) − g_{j}y_{2}P_{2}(θ).

Now clearly,

∑_{i=0}^{d} | \begin{matrix}f_{j} & g_{j} \\ f_{i} & g_{i} \end{matrix} | Ψt^{i} = Ψ | \begin{matrix}f_{j} & g_{j} \\ f(t) & g(t) \end{matrix} | ,

so that:

Ψ | \begin{matrix}f_{j} & g_{j} \\ f(t) & g(t) \end{matrix} | ∈ (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 × (m + 1)-matrix ( \begin{matrix}f_{0} … f_{m} \\ g_{0} … g_{m} \end{matrix} ) equals 2 and hence Ψf(t), Ψg(t) ∈ (U_{1}, U_{2}). Since (t − 1)Ψ = Ψt, it follows that Ψh(t) ∈ (U_{1}, U_{2}), for any h(t) ∈ (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))Ψ ∈ (U_{1}, U_{2}). The proof is complete. □

Corollary 10.5. Suppose that gcd(σ(U_{1}), σ(U_{2})) is a power of x_{1}z_{1} + x_{2}z_{2}. Then the hypergeometric system (28) is holonomic if and only if R(f(t), g(t)) ≠ 0.

Proof. Suppose that R(f(t), g(t)) = 0 and let ζ ∈ ℂ be a common root of the polynomials f, g. Since for any smooth univariate function h the product y_{2}^{ζ}h(y_{1}/y_{2}) is annihilated by the operator t − ζ = θ_{1} + θ_{2} − ζ, it follows that the space of analytic solutions to (28) has infinite dimension. It is known that a holonomic system can only have finitely many linearly independent solutions and hence (28) is not holonomic in this case.

On the other hand, if R(f(t), g(t)) ≠ 0, then by Lemma 10.4 the operator Ψ lies in the ideal (U_{1}, U_{2}). By the assumption of the corollary the principal symbols of U_{1}, U_{2} and Ψ are relatively prime and hence the system (28) is holonomic. □

Example 10.6. Consider the system quoted in the introduction, given by the two hypergeometric operators

H_{1} = x(θ_{x} +θ_{y} +a)(θ_{x} +b)−θ_{x}(θ_{x} +θ_{y} +c−1), H_{2} = y(θ_{x} +θ_{y} +a)(θ_{y} +b′)−θ_{y}(θ_{x} +θ_{y} +c−1)

for Appell’s function F_{1}. The operator Ψ in Lemma 10.3 equals in this case

Ψ = (x y) Ψ′ , where Ψ′ = (x − y)∂_{x}∂_{y} − b′∂_{x} + b∂_{y}.

When a − c + 1 ≠ 0, we deduce from Lemma 10.4 that (x y) Ψ′ lies in the D-ideal ⟨H_{1}, H_{2}⟩. In particular, all holomorphic solutions φ of the Appell system will also satisfy Ψ′(φ) = 0. We point out that some authors add this third equation to the system (cf. for instance [26, Page 48]). In fact, having this operator, the holonomicity of the system follows immediately.

We are now in a position to complete the proof of Theorem 10.1.

Proof of Theorem 10.1. Suppose that the polynomial gcd(σ(H_{1}), σ(H_{2})) vanishes along the hypersurface ay_{1}z_{1} + by_{2}z_{2} = 0. We aim to construct an operator in the ideal (H_{1}, H_{2}) whose principal symbol is not divisible by ay_{1}z_{1} + by_{2}z_{2}. The change of variables ξ_{1} = y_{1}^{1/a}, ξ_{2} = y_{2}^{1/b} transforms the operator aθ_{y_{1}} + bθ_{y_{2}} into the operator θ_{ξ_{1}} + θ_{ξ_{2}} and the system (23) into the system generated by the operators

(29)  \begin{matrix}Q̂_{1}(θ_{ξ_{1}}, θ_{ξ_{2}}) − ξ_{1}^{a} P̂_{1}(θ_{ξ_{1}}, θ_{ξ_{2}}), \\ Q̂_{2}(θ_{ξ_{1}}, θ_{ξ_{2}}) − ξ_{2}^{b} P̂_{2}(θ_{ξ_{1}}, θ_{ξ_{2}}), \end{matrix}

where P̂_{i}(u, v) = P_{i}(u/a, v/b), Q̂_{i}(u, v) = Q_{i}(u/a, v/b).

Let us introduce operators λ_{ia}^{k}, μ_{ia}^{k} acting on a bivariate polynomial P as follows:

(30)  λ_{ia}^{k}(P) = ∏_{j=1}^{k}(E_{i}^{−ja}P),  μ_{ia}^{k}(P) = ∏_{j=0}^{k−1}(E_{i}^{ja}P).

(Notice that the upper index here is not a power.) The next Weyl algebra identities follow directly from the definition of λ_{ia}^{k}, μ_{ia}^{k} (the arguments of all of the involved polynomials being θ_{ξ_{1}}, θ_{ξ_{2}}):

(31)  \begin{aligned}λ_{1a}^{k} (Q̂_{1})ξ_{1}^{a} & = & ξ_{1}^{a} Q̂_{1} λ_{1a}^{k−1} (Q̂_{1}), \\ λ_{2b}^{k} (Q̂_{2})ξ_{2}^{b} & = & ξ_{2}^{b} Q̂_{2} λ_{2b}^{k−1} (Q̂_{2}), \\ μ_{1a}^{k} (P̂_{1})ξ_{1}^{a} P̂_{1} & = & ξ_{1}^{a} μ_{1a}^{k+1} (P̂_{1}), \\ μ_{2b}^{k} (P̂_{2})ξ_{2}^{b} P̂_{2} & = & ξ_{2}^{b} μ_{2b}^{k+1} (P̂_{2}). \end{aligned}

Using (31) we arrive at the equalities

(32) (∑_{ν=0}^{b−1} ξ_{1}^{νa}λ_{1a}^{b−1−ν}(Q̂_{1})(θ_{ξ})μ_{1a}^{ν}(P̂_{1})(θ_{ξ})) (Q̂_{1}(θ_{ξ})−ξ_{1}^{a}P̂_{1}(θ_{ξ})) =
Q̂_{1}(θ_{ξ})λ_{1a}^{b−1}(Q̂_{1})(θ_{ξ}) − ξ_{1}^{ab}μ_{1a}^{b}(P̂_{1})(θ_{ξ}),

(33) (∑_{ν=0}^{a−1} ξ_{2}^{νb}λ_{2b}^{a−1−ν}(Q̂_{2})(θ_{ξ})μ_{2b}^{ν}(P̂_{2})(θ_{ξ})) (Q̂_{2}(θ_{ξ})−ξ_{2}^{b}P̂_{2}(θ_{ξ})) =
Q̂_{2}(θ_{ξ})λ_{2b}^{a−1}(Q̂_{2})(θ_{ξ}) − ξ_{2}^{ab}μ_{2b}^{a}(P̂_{2})(θ_{ξ}).

The differential operators (32) and (33) are Horn-type hypergeometric operators in the variables η_{1} = ξ_{1}^{ab} and η_{2} = ξ_{2}^{ab}. Let us write these operators in the form

\begin{matrix}Ũ_{1} = f(τ)Q̃_{1}(θ_{η}) − η_{1}g(τ)P̃_{1}(θ_{η}), \\ Ũ_{2} = f(τ)Q̃_{2}(θ_{η}) − η_{2}g(τ)P̃_{2}(θ_{η}), \end{matrix}

where f, g are univariate polynomials, τ = θ_{η_{1}} + θ_{η_{2}} and none of the principal symbols of the operators P̃_{i}(θ_{η}), Q̃_{i}(θ_{η}) vanish along the hypersurface η_{1}z_{1} + η_{2}z_{2} = 0. The existence of such polynomials f, g follows from the compatibility condition which is satisfied by (32),(33).

By Lemma 10.4 the operator Ψ̃ = η_{1}Q̃_{2}(θ_{η})P̃_{1}(θ_{η})−η_{2}Q̃_{1}(θ_{η})P̃_{2}(θ_{η}) lies in the ideal (Ũ_{1}, Ũ_{2}) as long as the parameters of the original Horn system (23) are generic. Notice that by construction the principal symbol of Ψ̃ does not vanish along the hypersurface η_{1}z_{1} + η_{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 ay_{1}z_{1} + by_{2}z_{2}. This completes the proof of Theorem 10.1. □

We are finally ready to show that Horn (B, c) is regular holonomic for generic c.

Theorem 10.7. Assume that all column sums of B are zero. For generic parameters c, the bivariate Horn system Horn (B, c) is regular holonomic.

Proof. By Theorem 10.1, the Horn system Horn (B, c) is holonomic. For generic parameter vectors c, we know that H_{B}(c) is regular holonomic, and Corollary 5.2 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. □

11. 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 · 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_{B}(c) is not deg(I) = d_{1} · d_{2}, unless we make the assumption that B has no linearly dependent rows in opposite open quadrants of ℤ^{2}.

If we follow the arguments that proved [26, Lemma 4.3.7], which is the main ingredient needed to prove that, when I_{A} is Cohen-Macaulay, rank (H_{A}(A · c)) = vol (A) for all c, we see that the crucial point is whether the n − m polynomials

(34)  ∑_{j=1}^{n} a_{ij}x_{j}z_{j} ∈ ℂ[x_{1}, …, x_{n}, z_{1}, …, z_{n}], i = 1, …, n − m,

form a regular sequence in ℂ(x_{1}, …, x_{n})[z_{1}, …, z_{n}]/I, where here we think of I as an ideal in the variables z_{1}, …, z_{n}. But if B has linearly dependent rows in opposite open quadrants, the polynomials (34) do not even form a homogeneous system of parameters!

Lemma 11.1. Let m = 2. If ⟨A · xz⟩ is ideal generated by the polynomials (34), then the ideal I + ⟨A · xz⟩ is artinian in ℂ(x_{1}, …, x_{n})[z_{1}, …, z_{n}]/I, if and only if B has no linearly dependent rows in opposite open quadrants of ℤ^{2}.

Proof. We need to investigate the intersection of the zero locus of ⟨A · xz⟩ 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 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 4.2.

The toric irreducible components of I we can deal with all at the same time: we know that ℂ(x)[z]/(I_{B} + ⟨A · xz⟩) is zero-dimensional. That just leaves the primary components of I corresponding to associated primes ⟨z_{i}, z_{j}⟩, 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 ⟨A · xz⟩ in an infinite set if and only if b_{i} and b_{j} are linearly dependent. □

As a consequence of Lemma 11.1 and the arguments in [26, Section 4.3], we have one case when the fact that I is a complete intersection will imply that the rank of H_{B}(c) does not depend on c.

Theorem 11.2. If B has no linearly dependent rows in opposite quadrants of ℤ^{2} then

rank (H_{B}(c)) = d_{1} · d_{2} for all c ∈ ℂ^{n}.

Notice that this result holds even when the rows of 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 [11].

There is another situation when we can apply the arguments from [26, Section 4.3] to prove that a certain holonomic rank does not depend on c. Let J be the ideal in ℂ[∂_{1}, …, ∂_{n}] obtained by saturating from I the components I_{ij} corresponding to linearly dependent rows of B. Then

deg(J) = d_{1} · d_{2} − ∑ ν_{ij},

where the sum runs over the linearly dependent rows of B that lie in opposite open quadrants of ℤ^{2}. As before, the methods in [26, Section 4.3] prove the following result.

Lemma 11.3. If J is Cohen-Macaulay,

rank (J + ⟨A · θ − A · c⟩) = deg(J).

The previous lemma and our rank formula for Horn systems have the following consequence.

Corollary 11.4. If J is Cohen-Macaulay and c is generic, the solution spaces of H_{B}(c) and J + ⟨A · θ − A · c⟩ coincide.

We believe that Corollary 11.4 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.

The natural question at this point is whether we can extend arguments in Section 9 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 [4]. Moreover, we used very strongly the fact that H_{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.

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 Björk 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.

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Dto. de Matemática, FCEyN, Universidad de Buenos Aires, (1428) Buenos Aires, Argentina.

  E-mail address: alidick@dm.uba.ar

Department of Mathematics, Harvard University,

Cambridge, MA 02138, USA.

  E-mail address: laura@math.harvard.edu

Department of Mathematics,

The University of Western Ontario,

London, Ontario N6A 5B7, Canada.

  E-mail address: tsadykov@uwo.ca