NONCONFLUENT HYPERGEOMETRIC FUNCTIONS

IN SEVERAL VARIABLES

AND THEIR SINGULARITIES

Mikael Passare, Timour Sadykov and August Tsikh

Abstract. The paper deals with singularities of nonconfluent hypergeometric functions in several variables. Typically such a function is a multi-valued analytic function with singularities along an algebraic hypersurface. We give a description of such hypersurfaces in terms of amoebas and the Newton polytopes of their defining polynomials. In particular we obtain a characterization of the zero set of the discriminant of a general algebraic equation in terms of amoebas.

Mathematics Subject Classification (1991): 32A05, 32A20, 33C70

1 Introduction

There exist several approaches to the notion of hypergeometric series, functions and systems of differential equations. In the present paper we use the definition of these objects which was introduced by Horn at the end of the 19th century [10]. His original definition of a hypergeometric series is particularly attractive because of its simplicity. A Laurent series in several variables is said to be hypergeometric if the quotient of its two adjacent coefficients depends rationally on the indices of summation.

In the present paper we study singularities of hypergeometric functions which are defined by means of analytic continuation of hypergeometric series. A hypergeometric series y(x) satisfies the so-called Horn hypergeometric system

x_{i}P_{i}(θ)y(x) = Q_{i}(θ)y(x), i = 1, …, n.  (1)

Here P_{i}, Q_{i} are nonzero polynomials depending on the vector differential operator θ = (θ_{1}, …, θ_{n}), θ_{i} = x_{i} \frac{∂}{∂x_{i}} . The nonconfluency of a hypergeometric series or the system (1) means that the polynomials P_{i} and Q_{i} are of the same degree:

deg P_{i} = deg Q_{i}, i = 1, ..., n.

These conditions can be expressed in terms of the Ore-Sato coefficient of a hypergeometric series satisfying the system (1) (see formulas (4) and (5)). Historically the Gauss hypergeometric differential equation was the first one to be studied in detail due to the remarkable fact that any linear homogeneous differential equation of order two with three regular singularities can be reduced to it. The singularities of the Gauss equation are 0, 1, ∞. The generalized ordinary hypergeometric differential equation which is a special case of the nonconfluent system (1) corresponding to n = 1 also has three singular points, namely 0, t, ∞, where t is the quotient of the coefficients in the leading terms in the polynomials P_{1} and Q_{1}. Thus the singular set of an ordinary hypergeometric differential equation is minimal in the following sense. There exist only two circular domains, namely {0 < |x| < |t|} and {|t| < |x| < ∞} in which any solution to the equation can be represented as a Laurent series (in the nonresonant case) or as a linear combination of the products of Laurent series and powers of log x (in the resonant case).

It turns out that algebraic singularities of the system of partial differential equations (1) enjoy a multidimensional analogue of this minimal property. It is convenient to formulate this property in the language of amoebas which were introduced by Gelfand, Kapranov and Zelevinsky in [9]. The amoeba of an algebraic set R = {R(x) = 0} is defined to be its image under the mapping Log : (x_{1}, ..., x_{n}) ↦ (log |x_{1}|, ..., log |x_{n}|). The complement of an amoeba consists of a finite number of convex connected components which correspond to domains of convergence of Laurent series expansions of single-valued functions with the singularities on R. The number of such components cannot be smaller than the number of vertices of the Newton polytope of the polynomial R(x). If these two numbers are equal then we say that the amoeba and the algebraic set R are minimal. In Section 5 we prove the following theorem. Theorem 7 Singularities of nonconfluent hypergeometric functions of finite branching are minimal.

This theorem implies that the amoeba of the discriminant of a general algebraic equation is minimal (Corollary 8).

The authors conjecture that the finite branching condition in Theorem 7 is unessential and that all A-discriminants have minimal amoebas (for definitions see Chapter 9 in [9]). This conjecture is to some extent supported by the results of Kapranov (see [12]).

Let us also mention the following results in the paper. Theorem 11 states that any meromorphic nonconfluent hypergeometric function is rational. In the last Section we study the problem of describing the class of rational hypergeometric functions. In the class of hypergeometric functions satisfying the Gelfand-Kapranov-Zelevinsky system of equations this problem was first considered in [2]. Theorem 12 gives a necessary condition for the Horn system to possess a rational solution. The statement of Proposition 14 emphasizes the fact that only very few rational functions are hypergeometric. The class of rational hypergeometric functions which is described in this Proposition consists of those which are contiguous to Bergman kernels of complex ellipsoidal domains.

The proofs of the main results in the paper use the notions of the support and the fan of a hypergeometric series, some facts from toric geometry and the two-sided Abel lemma which is proved in Section 6. Recall that the usual (one-sided) Abel lemma (see [8] or [13]) gives the following relation between the domain of convergence of a Puiseux series and its support (i.e. the set of summation).

Lemma 1 (Abel’s lemma for Puiseux series) Let y(x) be a Puiseux series with a nonempty domain of convergence D. For any x^{(0)} ∈ D and any cone C containing the convex hull of the support of y(x) we have Log(x^{(0)}) − C^{∨} ⊂ Log(D). Here C^{∨} is the dual cone to C.

The two-sided Abel lemma for hypergeometric Puiseux series states that the domain Log (D) is itself contained in a suitable translation of the cone −C^{∨}.

The authors are thankful to A. Dickenstein for fruitful discussions. Passare and Tsikh were supported by a grant from the Swedish Royal Academy of Sciences. Sadykov and Tsikh were supported by the Russian Foundation for Basic Research, grant 00-15-96140. Tsikh is grateful to Max-Planck-Institut für Mathematik in Bonn for its hospitality and financial support.

2 Some basic notations and definitions

To study the singularities of solutions to the Horn system (1) we consider the characteristic variety of this system. Let D denote the Weyl algebra of differential operators with polynomial coefficients in n variables [1]. For any differential operator P ∈ D, P = ∑_{|α|≤m} c_{α}(x)(\frac{∂}{∂x})^{α} its principal symbol σ(P)(x, z) ∈ ℂ[x_{1}, …, x_{n}, z_{1}, …, z_{n}] is defined by

σ(P)(x, z) = ∑_{|α|=m} c_{α}(x)z^{α}.

We denote by G_{i} the differential operator x_{i}P_{i}(θ) − Q_{i}(θ) in the ith equation of the Horn system (1). Let M = D/ ∑_{i=1}^{n} DG_{i} be the left D-module associated with the system (1) and let J ⊂ D denote the left ideal generated by the differential operators G_{1}, …, G_{n}. By definition (see [1], Chapter 5, § 2) the characteristic variety char(M) of the Horn system is given by

char(M) = {(x, z) ∈ ℂ^{2n} : σ(P)(x, z) = 0, for all P ∈ J}.

We define the set U_{M} ⊂ ℂ^{n} by

U_{M} = {x ∈ ℂ^{n} : ∃ z ≠ 0 such that (x, z) ∈ char(M)}.

It follows from Proposition 8.1.3 and Theorem 8.3.1 in [11] and Theorem 7.1 in Chapter 5 of [1] that a solution to (1) can only be singular on U_{M}. Since any equation of the form σ(P)(x, z) = 0 is homogeneous in z, it follows that U_{M} is the image of char(M) under the projection of the direct product ℂ^{n}×ℙ^{n−1} → ℂ^{n} onto its first factor. Using the main theorem of elimination theory (see § 2C in [17]) one can conclude that this image is an algebraic set, possibly the whole of ℂ^{n}. In the latter case the singularities of a solution to the Horn system are not necessarily algebraic. For instance, if every differential operator G_{i} contains the factor (θ_{1} + ... + θ_{n}) then any sufficiently smooth function depending on the quotients \frac{x_{1}}{x_{n}} , ..., \frac{x_{n−1}}{x_{n}} is a solution to the system (1).

In the present paper we consider systems of the Horn type which satisfy the condition U_{M} ≠ ℂ^{n}. In this case U_{M} is a proper algebraic subset of ℂ^{n}. Its irreducible components of codimension greater than one are removable as long as we are concerned with holomorphic solutions to the Horn system. Thus the singular set of a solution to (1) is algebraic and it is contained in the union of irreducible components of codimension one. We denote this union by R and call it the singular set of the Horn system. Let R(x) be the defining function of the set R, i.e.

R = {R(x) = 0}.

The polynomial R(x) will be referred to as the resultant of the Horn system (1). To find a polynomial whose zero set is R is a difficult task which requires the full use of elimination theory. There exists however a simple special case when the set R can be embedded into the zero set of some polynomial which one can algorithmically compute. Let H_{i}(x, z) be the principal symbol of the differential operator G_{i} in the ith equation of the Horn system (1). Since the polynomials H_{1}, ..., H_{n} are homogeneous in z_{1}, ..., z_{n} they determine the classical resultant R[H_{1}, ..., H_{n}] which is a polynomial in x_{1}, ..., x_{n} (see [9], Chapter 13). For the convenience of future references we formulate the following simple proposition.

Proposition 2 The singular set R of the Horn system (1) lies in the zero set of the resultant R[H_{1}, ..., H_{n}] of the principal symbols of the operators in (1).

To prove this Proposition it suffices to notice that for x^{(0)} ∈ U_{M} the system of equations H_{1}(x^{(0)}, z) = … = H_{n}(x^{(0)}, z) = 0 (considered as a system of algebraic equations in z_{1}, …, z_{n} whose coefficients depend on x^{(0)}) has a solution in ℂ^{n} \ {0}. This yields that the resultant of the homogeneous forms H_{1}(x, z), …, H_{n}(x, z) with respect to the variables z_{1}, …, z_{n} vanishes at x^{(0)} (see [9], Chapter 13). Thus the singular locus of a solution to the Horn system (1) is contained in the zero set of the resultant R[H_{1}, ..., H_{n}]. Notice that the vanishing of this resultant at a point x^{(0)} ∈ ℂ^{n} is equivalent to the condition that the sequence of the principal symbols {H_{i}(x^{(0)}, z)}_{i=1}^{n} is not regular in the polynomial ring ℂ[z_{1}, …, z_{n}].

3 Puiseux series solutions to the Horn system and their supports

The Horn system (1) as well as the Gelfand-Kapranov-Zelevinsky system (see [8]) has the remarkable property that under some natural assumptions there exists a basis in the space of its holomorphic solutions consisting of (Puiseux) series with the center at the origin (see [8] for the Gelfand-Kapranov-Zelevinsky system and [19] for the Horn system). In this Section we introduce some terminology and present preliminary results which will be used later for describing the singular set of the Horn system.

Suppose that a formal Puiseux series centered at the origin satisfies the Horn system (1). Such a series can be written as a linear combination of formal shifted Laurent series, i.e. series of the form

y(x) = x^{γ} ∑_{s∈ℤ^{n}} φ(s)x^{s}.  (2)

Here the shift is determined by the initial monomial x^{γ}, γ = (γ_{1}, …, γ_{n}) ∈ ℂ^{n}, Re γ_{i} ∈ [0, 1), x^{s} = x_{1}^{s_{1}} … x_{n}^{s_{n}}. Suppose that the series (2) is a solution to (1). Computing the action of the operator x_{i}P_{i}(θ) − Q_{i}(θ) on this series we arrive at the system of difference equations

φ(s + e_{i})Q_{i}(s + γ + e_{i}) = φ(s)P_{i}(s + γ), i = 1, …, n,  (3)

where {e_{i}}_{i=1}^{n} is the standard basis of ℤ^{n}. The system (3) is equivalent to (1) as long as we are concerned with those solutions to the Horn system which admit a series expansion of the form (2).

The system of difference equations (3) is in general not solvable without further restrictions on P_{i}, Q_{i}. Let R_{i}(s) denote the rational function P_{i}(s)/Q_{i}(s + e_{i}), i = 1, ..., n. Increasing the argument s in the ith equation of (3) by e_{j} and multiplying the obtained equality by the jth equation of (3), we arrive at the relation φ(s + e_{i} + e_{j})/φ(s) = R_{i}(s+e_{j})R_{j}(s). Similarly φ(s+e_{i} +e_{j})/φ(s) = R_{j}(s+e_{i})R_{i}(s). Thus the conditions R_{i}(s + e_{j})R_{j}(s) = R_{j}(s + e_{i})R_{i}(s), i, j = 1, ..., n are in general necessary for (3) to be solvable. Throughout this paper we assume that the polynomials P_{i}, Q_{i} defining the Horn system (1) satisfy these relations.

Since the polynomials P_{i}, Q_{i} are assumed to be representable as products of linear factors, it follows by the Ore-Sato theorem (see [20] and [7], § 1.2) that the general solution to the system of difference equations (3) is of the form

φ(s) = t_{1}^{s_{1}} … t_{n}^{s_{n}} ∏_{i=1}^{p} Γ(⟨A_{i}, s + γ⟩ − c_{i})ϕ(s).  (4)

Here t_{i}, c_{i} ∈ ℂ, A_{i} = (A_{i1}, …, A_{in}) ∈ ℤ^{n}, p ∈ ℕ_{0}, ϕ(s) is an arbitrary periodic function with the period 1 in each variable. The fact that all the Γ-functions in (4) are in the numerator is unessential: using the identity Γ(z)Γ(1 − z) = π/ sin πz and choosing the periodic function ϕ(s) in an appropriate way (see [19]), one can move them into the denominator. A formal series (2) with the coefficient (4) is called a formal solution to the system (1). We will call any expression of the form (4) the Ore-Sato coefficient of a hypergeometric series (or of the system (1)).

Remark 1 Conversely, the Ore-Sato coefficient (4) defines the system (1) in the sense that for any i = 1, …, n the quotient φ(s + e_{i})/φ(s) equals P_{i}(s)/Q_{i}(s + e_{i}). For instance, the Ore-Sato coefficient (12) in Example 1 (see below) defines the Horn system (10).

The specific form of (4) corresponds to our assumption that the polynomials P_{i}, Q_{i} can be represented as products of linear factors. In general an Ore-Sato coefficient can include a rational function which is not factorizable up to linear factors (see [7], § 1.2).

One can easily check that in terms of the parameters of the Ore-Sato coefficient φ(s) the nonconfluency condition deg P_{i} = deg Q_{i} can be written in the form

∑_{i=1}^{p} A_{i} = 0.  (5)

Recall that in this paper we only deal with nonconfluent hypergeometric series.

Any shifted Laurent series solution to (1) (formal as well as convergent) can be written in the form

y(x) = x^{γ} ∑_{s∈S} φ(s)x^{s},  (6)

where φ(s) is given by (4) and S is a subset of ℤ^{n} on which φ(s) ≠ 0. The set S will be called the support of the series (6). The support S is called irreducible if there exists no series solution to (1) supported in a proper nonempty subset of S. A set S ⊂ ℤ^{n} is said to be ℤ^{n}-connected if any two points of S can be connected by a polygonal line with unit sides and vertices in S.

Proposition 3 (see below) describes all possible supports of (formal) series solutions to (1) and Proposition 5 allows one to find those of them which have nonempty domains of convergence. While looking for a solution to (3) which is different from zero on some subset S of ℤ^{n} we will assume that the polynomials P_{i}(s), Q_{i}(s), the set S and the vector γ satisfy the condition

|P_{i}(s + γ)| + |Q_{i}(s + γ + e_{i})| ≠ 0,  (7)

for any s ∈ S and for all i = 1, …, n. This assumption eliminates the case when a solution to (3) can independently take arbitrary values at two adjacent points in the set S. The following statement (see [19]) gives necessary and sufficient conditions for a solution to the system (3) supported in some set S ⊂ ℤ^{n} to exist.

Proposition 3 (Sadykov [19]) For S ⊂ ℤ^{n} define

Si^{′} = {s ∈ S : s + e_{i} ∉ S}, Si^{′′} = {s ∉ S : s + e_{i} ∈ S}, i = 1, …, n.

Suppose that the conditions (7) are satisfied on S. Then there exists a solution to the system (3) supported in S if and only if the following conditions are fulfilled:

P_{i}(s + γ)|_{S′_{i}} = 0, Q_{i}(s + γ + e_{i})|_{S′′_{i}} = 0, i = 1, …, n,  (8)

P_{i}(s + γ)|_{S\S′_{i}} ≠ 0, Q_{i}(s + γ + e_{i})|_{S} ≠ 0, i = 1, …, n.  (9)

By the definition the union of the sets Si^{′}, Si^{′′}, i = 1, ..., n is a discrete analogue of the boundary of the set S. Since the polynomials P_{i}, Q_{i} are assumed to be representable as products of linear factors, it follows from (8) that Si^{′} and Si^{′′} lie on hyperplanes. The conditions (9) yield that these hyperplanes bound the set S. Thus we can formulate the following result.

Proposition 4 The convex hull of the support of a series solution to the Horn system is a polyhedral set.

Example 1 Let us consider the following system of partial differential equations of the Horn type

\{ \begin{matrix}x_{1}(θ_{1} + θ_{2})(θ_{1} − 2)y(x) & = (θ_{1} − 1)(θ_{1} − 4)y(x), \\ x_{2}(θ_{1} + θ_{2})(θ_{2} − 3)y(x) & = (θ_{2} − 1)(θ_{2} − 5)y(x). \end{matrix}  (10)

Assuming that y(x) admits a Laurent series expansion (2) with γ = 0, we arrive at the system of difference equations

\{ \begin{matrix}φ(s + e_{1})s_{1}(s_{1} − 3) = & φ(s)(s_{1} + s_{2})(s_{1} − 2), \\ φ(s + e_{2})s_{2}(s_{2} − 4) = & φ(s)(s_{1} + s_{2})(s_{2} − 3). \end{matrix}  (11)

In accordance with the Ore-Sato theorem (see [20] and [7], § 1.2) the general solution to the system (11) is given by the function

φ(s) = (s_{1} − 3)(s_{2} − 4)\frac{Γ(s_{1} + s_{2})}{Γ(s_{1})Γ(s_{2})}ϕ(s),  (12)

where ϕ(s) is an arbitrary periodic function with the period 1 in s_{1} and s_{2}. There exist eight ℤ^{2}-connected subsets of the lattice ℤ^{2} which satisfy the conditions of Proposition 3, namely

\begin{aligned}S_{1} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 1 ≤ s_{1} ≤ 2, 1 ≤ s_{2} ≤ 3\}, \\ S_{2} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 4 ≤ s_{1}, 5 ≤ s_{2}\}, \\ S_{3} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 5 ≤ s_{2}, s_{1} + s_{2} ≤ 0\}, \\ S_{4} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 4 ≤ s_{1}, s_{1} + s_{2} ≤ 0\}, \\ S_{5} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 4 ≤ s_{1}, 1 ≤ s_{2} ≤ 3\}, \\ S_{6} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : s_{1} + s_{2} ≤ 0, 1 ≤ s_{2} ≤ 3\}, \\ S_{7} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 1 ≤ s_{1} ≤ 2, 5 ≤ s_{2}\}, \\ S_{8} & = \{(s_{1}, s_{2}) ∈ ℤ^{2} : 1 ≤ s_{1} ≤ 2, s_{1} + s_{2} ≤ 0\}. \end{aligned}

These irreducible supports of solutions to (11) are displayed in Figure 1.

[figure: s_{1}; s_{2}; S_{1}; S_{2}; S_{3}; S_{4}; S_{5}; S_{6}; S_{7}; S_{8}]

Fig. 1 The irreducible supports of the solutions to the Horn system (10)

[figure]

Fig. 2 The Newton polytope of the resultant of (10)

[figure]

Fig. 3 The fan of the Horn system (10) (see Section 4)

Using the formula (12) in [19] for defining the periodic function ϕ(s), one can compute the sums of the corresponding Laurent series. Let y_{i}(x) denote the series solution to (10) with the support S_{i}. These functions are defined up to unessential constant factors which we choose in a specific way in order to make the formulas simpler. Computations (which were performed in MAPLE) show that

y_{1}(x) = 3x_{1}x_{2} + 4x_{1}x_{2}^{2} + 3x_{1}x_{2}^{3} + 3x_{1}^{2}x_{2} + 6x_{1}^{2}x_{2}^{2} + 6x_{1}^{2}x_{2}^{3},
y_{5}(x) = x_{1}^{4}x_{2}(6x_{1}^{3}x_{2}^{2} + 6x_{1}^{3}x_{2} − 27x_{1}^{2}x_{2}^{2} + 3x_{1}^{3} − 26x_{1}^{2}x_{2} +
45x_{1}x_{2}^{2} − 12x_{1}^{2} + 40x_{1}x_{2} − 30x_{2}^{2} + 15x_{1} − 20x_{2} − 6)/(1 − x_{1})^{5},

y_{7}(x) = \frac{x_{1}x_{2}^{5}(6x_{1}x_{2}^{2} − 18x_{1}x_{2} + 3x_{2}^{2} + 15x_{1} − 8x_{2} + 5)}{(1 − x_{2})^{4}},

y_{2}(x) = \frac{x_{1}x_{2}(6x_{1}^{2} + 14x_{1}x_{2} + 5x_{2}^{2} − 9x_{1} − 8x_{2} + 3)}{(1 − x_{1} − x_{2})^{4}} − y_{1}(x) − y_{5}(x) + y_{7}(x)

(we omit an explicit but cumbersome formula for y_{2}(x)). The series supported in S_{2}, S_{3}, S_{4} represent the same solution to our system since they represent the same rational function in different domains. Finally, y_{6}(x) = y_{1}(x) + y_{5}(x) and y_{8}(x) = y_{1}(x)+y_{7}(x). It follows from Theorem 8 in [19] that the space of holomorphic solutions to the system (10) has dimension 4 at any point x ∈ ℂ^{2} such that (1 − x_{1})(1 − x_{2})(1 − x_{1} − x_{2}) ≠ 0. Hence the rational functions y_{1}(x), y_{2}(x), y_{5}(x), y_{7}(x) form a basis in this space. Notice that the resultant of the principal symbols of the operators in the system (10) is given by the polynomial (x_{1}x_{2})^{4}(1 − x_{1})(1 − x_{2})(1 − x_{1} − x_{2}).

To conclude this Section we formulate one more statement on the properties of supports of hypergeometric series which will be used in the sequel.

Proposition 5 A nonconfluent hypergeometric series with the support S has a nonempty domain of convergence if and only if the convex hull of S is a polyhedral set which is contained in a translation of a strongly convex cone. The domain of convergence of the series (6) is independent on the parameters c_{1}, ..., c_{p} in the formula (4) (we disregard exceptional values of these parameters for which (6) terminates or reduces to a linear combination of hypergeometric series in fewer variables).

The first conclusion of this proposition follows from Proposition 4, the Lemma in § 4.1 of [7] and the properties of hypergeometric series in one variable (see Chapter 1 in [22]). The second conclusion of the proposition follows from Theorem 1 in § 4.1 of [22].

Finally we remark that there exists a simple relation between the domain of convergence of a nonconfluent hypergeometric series and its support. This relation is described by the two-sided Abel lemma which is proved in Section 6.

4 The fan of the Horn system

By an affine convex cone we mean a set of the form C + ξ, where C is a convex cone in ℝ^{n} with apex at the origin and ξ ∈ ℝ^{n}. Let C_{1} + ξ_{1} and C_{2} + ξ_{2} be affine convex cones with C_{1}, C_{2} being convex cones and ξ_{1}, ξ_{2} ∈ ℝ^{n}. We say that C_{1} + ξ_{1} is smaller than C_{2} + ξ_{2} if C_{1} ⊂ C_{2}. If C_{1} = C_{2} then the corresponding affine cones are said to be equal. For a convex set B ⊂ ℝ^{n} its recession cone C_{B} is defined to be C_{B} = {s ∈ ℝ^{n} : u + λs ∈ B, ∀u ∈ B, λ ≥ 0} (see [25], Chapter 1). That is, the recession cone of a convex set is the maximal element in the family of those cones whose shifts are contained in this set.

For the reason of brevity the recession cone of the convex hull of the support of a Puiseux series solution to the Horn system will be referred to as the cone of its support. It has nonempty interior if and only if the corresponding hypergeometric series cannot be represented as a linear combination of hypergeometric series in fewer variables which depend monomially on the original ones. In Example 1 the cone of the irreducible support S_{2} is the positive quadrant, the cone of S_{5} is {(s_{1}, s_{2}) : s_{1} ≥ 0, s_{2} = 0}, the cone of S_{1} is the origin.

Here and later we assume that the rank of the matrix with the rows A_{1}, ..., A_{p} is n since otherwise the series with the coefficient (4) can be reduced to a hypergeometric series in fewer variables. Let I = (i_{1}, ..., i_{n}), i_{j} ∈ {1, ..., p} be a multi-index such that the vectors A_{i_{1}}, ..., A_{i_{n}} are linearly independent. Let γ_{I} be the solution of the system of linear equations ⟨A_{i_{j}}, s⟩ − c_{i_{j}} = 0, j = 1, ..., n and define the set K_{I} by K_{I} = {s ∈ ℤ^{n} : (⟨A_{i_{j}}, s + γ_{I}⟩ − c_{i_{j}}) ≤ 0, j = 1, ..., n}. Let ℤ^{n} + γ denote the shift in ℂ^{n} of the lattice ℤ^{n} with respect to the vector γ.

Definition 1 We say that the parameter c = (c_{1}, ..., c_{p}) ∈ ℂ^{p} is generic, if for any multi-index I as above none of the hyperplanes ⟨A_{j}, s + γ_{I}⟩ − c_{j} = 0, j ∉ {i_{1}, ..., i_{n}} meets the shifted lattice ℤ^{n} + γ_{I}.

Proposition 6 If the vector c = (c_{1}, ..., c_{p}) is generic then there exists a one-to-one correspondence between the cones of the supports of the convergent series solutions to the Horn system of the form (6) and the multi-indices I = (i_{1}, ..., i_{n}) such that the vectors A_{i_{1}}, ..., A_{i_{n}} are linearly independent. The recession cone of the convex hull of the support of any such series is strongly convex and polyhedral.

Proof. For a multi-index I as above consider the shifted Laurent series

y_{I}(x) = ∑_{s∈K_{I}} t^{s} ∏_{i=1}^{p} Γ(⟨A_{i}, s + γ_{I}⟩ − c_{i})x^{s+γ_{I}}.  (13)

Since the parameter c is assumed to be generic, it follows from Proposition 3 that the coefficient of the series (13) satisfies the equations (3) everywhere on ℤ^{n}, i.e. that (13) is at least a formal solution to the Horn system (1). By Proposition 5 the series (13) has nonempty domain of convergence since its support is contained in a strongly convex (and simplicial) affine cone. Thus with any multi-index I as above one can associate the cone C_{I} of the support of the series (13).

It follows by Proposition 3 that if the support of a formal series solution to (1) meets at most n−1 linearly independent hyperplanes of the form ⟨A_{j}, s+γ⟩−c_{j} = 0 for some γ ∈ ℂ^{n} then it cannot be contained in any strongly convex affine cone and by Proposition 5 the series is divergent. By the assumption the parameter c is generic and hence the support of such a series cannot meet more than n hyperplanes of this form. If it meets exactly n hyperplanes with the linearly independent normals A_{i_{1}}, ..., A_{i_{n}} then the cone of the support of this series must coincide with C_{I} since it is bounded by the same hyperplanes. Thus the correspondence between linearly independent subsets of the set of vectors {A_{1}, ..., A_{p}} and the cones of the supports of shifted Laurent series solutions to (1) is one-to-one. The claim about the recession cone of the convex hull of the support of y_{I}(x) follows from Proposition 1.12 in [25] since the convex hull of K_{I} is a strongly convex affine polyhedral cone. □

Remark 2 Proposition 6 shows that adding new elements to the family of vectors {A_{i}}_{i=1}^{p} can only increase the number of solutions to the Horn system which is defined by the Ore-Sato coefficient (4) as long as the vector c remains generic.

We now associate with a nonconfluent Horn system a set of strongly convex polyhedral cones which will play an important role in the sequel. Recall that for a cone C ⊂ ℝ^{n} its dual is defined by C^{∨} = {v ∈ ℝ^{n} : ⟨u, v⟩ ≥ 0, ∀u ∈ C}. For any multi-index I = (i_{1}, ..., i_{n}) such that the vectors A_{i_{1}}, ..., A_{i_{n}} are linearly independent we denote by C_{I} the recession cone of the convex hull of the set K_{I} whose shift supports the series (13). We partially order the finite family {C_{I}} of strongly convex polyhedral cones with respect to inclusion and denote the maximal elements by C_{I^{(1)}}, ..., C_{I^{(d)}}. Let us introduce the cones B_{j} = −C_{I^{(j)}}^{∨}, j = 1, ..., d. Since for any I as above the polyhedral cone C_{I} has a nonempty interior, it follows that B_{j} is a strongly convex polyhedral cone. The nonconfluency condition (5) implies that ⋃_{j=1}^{d} B_{j} = ℝ^{n}. If the cones B_{1}, ..., B_{d} can be identified with the set of the maximal cones of some complete fan then we call it the fan of the Horn system (1).

If n = 2 then {B_{j}}_{j=1}^{d} is always the set of the maximal cones of some complete fan. For n ≥ 3 this is not necessarily the case. For instance, let n = 3 and let A_{1} = (1, 0, 0), A_{2} = (0, 1, 0), A_{3} = (0, 0, 2), A_{4} = (−1, 0, −1), A_{5} = (0, −1, −1). The multi-indices I^{(1)} = (1, 4, 5) and I^{(2)} = (2, 4, 5) define maximal cones but the intersection of their duals has a nonempty interior.

5 Minimality of the singularities of hypergeometric functions and discriminants

As we have already mentioned in the Introduction, the singular set of a hypergeometric function in one variable is minimal. In this Section we will prove that multivariate rational hypergeometric functions enjoy an analogous property. It turns out to be convenient to express this property using the notion of amoebas which was introduced by Gelfand et al. in [9] (see Chapter 6, § 1). The amoeba A_{f} of a Laurent polynomial f(x) (or of the algebraic hypersurface f(x) = 0) is defined to be the image of the hypersurface f^{−1}(0) under the map Log : (x_{1}, ..., x_{n}) ↦ (log |x_{1}|, ..., log |x_{n}|). This name is motivated by the typical shape of A_{f} with tentacle-like asymptotes going off to infinity (see Figure 5). We quote the following result from [9] (see Chapter 6, Corollary 1.6), which describes the connection between the amoeba of a Laurent polynomial f and Laurent series developments of 1/f. Theorem A (Gelfand, Kapranov, Zelevinsky [9]) The connected components of the amoeba complement ^{c}A_{f} are convex, and they are in bijective correspondence with the different Laurent series expansions centered at the origin of the rational function 1/f.

Recall that the Newton polytope N_{f} of a Laurent polynomial f is defined to be the convex hull in ℝ^{n} of the support of f. The following result shows that the Newton polytope N_{f} reflects the structure of the amoeba A_{f} (see Theorem 2.8 and Proposition 2.6 in [4]). Theorem B (Forsberg, Passare, Tsikh [4]) Let f be a Laurent polynomial and let {M} denote the family of connected components of the amoeba complement ^{c}A_{f}. There exists an injective function ν : {M} → ℤ^{n} ∩ N_{f} such that the cone which is dual to N_{f} at the point ν(M) coincides with the recession cone of M.

The cited theorems imply that the number of Laurent series expansions with the center at the origin of the rational function 1/f is at least equal to the number of vertices of the Newton polytope N_{f} and at most equal to the number of integer points in N_{f}. Varying the coefficients of the Laurent polynomial f with the fixed Newton polytope N_{f}, one can attain the upper (see [15]) as well as the lower (see [18]) bounds for the number of connected components of^{c}A_{f}. Moreover, the vertices of the Newton polytope are always assumed by the function ν (see [16]) and by Theorem B the recession cones of those connected components of ^{c}A_{f} which correspond to the vertices of N_{f} have nonempty interior.

In this Section we show that the number of Laurent series expansions with the center at the origin of a rational hypergeometric function is always equal to the number of vertices of the Newton polytope of its denominator. For the sake of brevity we use the following definition.

Definition 2 The amoeba A_{f} of a Laurent polynomial f (or, equivalently, the algebraic hypersurface f(x) = 0) is called minimal if the number of connected components of the amoeba complement ^{c}A_{f} equals the number of vertices of the Newton polytope N_{f}.

In view of Theorem B it is obvious that the amoeba A_{f} is minimal if and only if the recession cone of every connected component of the set ^{c}A_{f} has a nonempty interior. The main observation in this Section is the following theorem.

Theorem 7 Singularities of nonconfluent hypergeometric functions of finite branching are minimal.

Proof. First of all we remark that a function of finite branching which is analytic in a domain of the form Log^{−1}M where M ⊂ ℝ^{n} is a convex set can be expanded into a Puiseux series in this domain. Indeed, since such a domain is homotopically equivalent to the torus (ℂ^{∗})^{k} for some k ≤ n, it follows that a suitable monomial change of variables makes the function single-valued in this domain. Expanding this new function into a Laurent series and going back to the old variables we obtain a Puiseux series expansion for the original function.

Let A be the amoeba of the singular hypersurface of a hypergeometric function y(x) of finite branching and let M ⊂ ^{c}A be a connected component of its complement. By the remark after Definition 2 it suffices to show that the recession cone C_{M} of the set M has a nonempty interior.

For ζ ∈ ∂M let Y_{ζ} ⊂ ℝ^{n} denote the half-space which is bounded by a supporting hyperplane of M at the point ζ and contains M. There exists a sequence of points {ζ_{i}}_{i=1}^{∞} ⊂ ∂M such that the recession cone of the set ⋂_{i=1}^{∞} Y_{ζ_{i}} coincides with C_{M}. Since A is the logarithmic image of the set of singularities of the function y(x), for any i ∈ ℕ there exists a germ G_{i} of y(x) which cannot be continued analytically through at least one point in the fiber Log^{−1}ζ_{i}. As we have remarked earlier, the analytic continuation of G_{i} into the domain Log^{−1}M can be expanded into a Puiseux series L_{i} whose domain of convergence contains Log^{−1}M. Let L^{(k)} = ∑_{i=1}^{k} L_{i}. The series L^{(k)} satisfies the same hypergeometric system of equations as y(x) since it is a linear combination of solutions to this system. By the construction L^{(k)} is not identically equal to zero. We denote the domain of convergence of the series L^{(k)} by Ω_{k}. By the construction M ⊂ Log Ω_{k} and hence Ω_{k} is nonempty. Moreover the recession cone C_{LogΩ_{k}} is a subset of the recession cone of the finite intersection ⋂_{i=1}^{k} Y_{ζ_{i}}.

Suppose that the cone C_{M} has the empty interior. The two-sided Abel lemma which will be proved in Section 6 states that for a nonconfluent hypergeometric Puiseux series L with the domain of convergence Ω one has C_{Log Ω} = −C_{L}^{∨}, where C_{L} is the cone of of the support of L and C_{Log Ω} is the recession cone of the set Log Ω. Thus we have −C_{L^{(k)}}^{∨} = C_{LogΩ_{k}} ⊂ C_{⋂_{i=1}^{k} Y_{ζ_{i}}} and hence the set ⋃_{k=1}^{∞} C_{L^{(k)}} is not strongly convex. By Proposition 4 the cone C_{L^{(k)}} is polyhedral with its boundary being a subset of the union of the zero sets of the polynomials P_{1}, ..., P_{n}, Q_{1}, ..., Q_{n}. Since this union is a finite arrangement of hyperplanes it follows that the family of cones {C_{L^{(k)}}}_{k=1}^{∞} can only contain a finite number of distinct elements. Therefore there exists m ∈ ℕ such that the cone C_{L^{(m)}} is not strongly convex. This contradicts the statement of Proposition 5 and completes the proof. □

Theorem 7 allows one to derive the following property of discriminants of algebraic equations.

Corollary 8 The amoeba of the discriminant of a general algebraic equation is minimal.

Proof. Any algebraic equation can be reduced to an equation of the form

y^{m} + x_{1}y^{m_{1}} + ... + x_{n}y^{m_{n}} − 1 = 0,  (14)

where m, m_{i} ∈ ℕ, m > m_{1} > ... > m_{n} ≥ 1, y is the unknown, x = (x_{1}, ..., x_{n}) ∈ ℂ^{n}. It was shown in [14] that the solution y(x) to (14) (which is considered as a multi-valued analytic function depending on x_{1}, ..., x_{n}) satisfies the system of partial differential equations

(−1)^{m_{i}}m^{m} \frac{∂^{m}y}{∂x_{i}^{m}} = ∏_{j=0}^{m_{i}−1}(m_{1}θ_{1} + ... + m_{n}θ_{n} + 1 + mj) ×

× ∏_{j=0}^{m′_{i}−1}(m1^{′}θ_{1} + ... + mn^{′}θ_{n} − 1 + mj)y, i = 1, ..., n,  (15)

where mi^{′} = m − m_{i}. The singular set of the function y(x) is the discriminant of the equation (14). Multiplying the ith equation of (15) with x_{i}^{m}, using the identity x_{i}^{m} \frac{∂^{m}}{∂x_{i}^{m}} = ∏_{j=0}^{m−1}(θ_{i} − j) and making the monomial change of variables ξ_{i} = x_{i}^{m} we reduce the Mellin system (15) to a system of the form (1). Thus y(ξ) is a nonconfluent hypergeometric function in the sense of Horn. Since the function y(x) is of finite branching, so is y(ξ). By Theorem 7 the singular set of y(ξ) is minimal. Since a monomial change of variables corresponds to a linear transformation of the amoeba space (see [4]), it follows that such a change of variables cannot affect the minimality of the singularity of y(x). Thus the amoeba of the discriminant of the algebraic equation (14) is minimal. □

The cubic equation is considered in detail in Example 3. The amoeba of the singular locus of a solution to the reduced system is displayed in Figure 5.

Theorem 7 implies in particular that the number of connected components of the complement of the amoeba of the singular hypersurface of a rational hypergeometric function equals the number of vertices of the Newton polytope of its denominator. It turns out that in some cases knowing the hypergeometric system which is satisfied by a given rational function allows one to compute the number of vertices of the Newton polytope of its denominator. We illustrate this fact by means of the following important family of rational hypergeometric functions which are defined as the Bergman kernels of complex ellipsoidal domains (see [5] and [26]). This family will be used in Section 7 for describing rational hypergeometric functions satisfying some systems of equations of the Horn type.

Consider the family of complex ellipsoidal domains defined by

D^{p_{1},... ,p_{n}} = {x ∈ ℂ^{n} : |x_{1}|^{2/p_{1}} + … + |x_{n}|^{2/p_{n}} < 1},

where p_{i} = 1, 2, 3, …, i = 1, …, n. The Bergman kernel K_{p_{1},... ,p_{n}}(x) for this domain was computed explicitly in [26]. It was shown that

K_{p_{1},... ,p_{n}}(x) = \frac{1}{π^{n}} ∑_{s∈ℕ_{0}^{n}} \frac{Γ(p_{1}(s_{1} + 1) + … + p_{n}(s_{n} + 1) + 1)}{∏_{i=1}^{n} p_{i}Γ(p_{i}(s_{i} + 1))}x^{s}.  (16)

The sum of this series is given by the function

K_{p_{1},... ,p_{n}}(x) = \frac{1}{π^{n}} \frac{1}{p_{1} … p_{n}} \frac{∂^{n}}{∂x_{1} … ∂x_{n}} ∑_{j_{1}=1}^{p_{1}} … ∑_{j_{n}=1}^{p_{n}} \frac{1}{1 − y_{j_{1}1} − … − y_{j_{n}n}} ,  (17)

where y_{j_{i}i} = x_{i}^{1/p_{i}}ε_{ji}, ε_{ji} are all the p_{i}-th roots of unity, j_{i} = 1, …, p_{i}, i = 1, …, n. The expression under the sign of the partial derivatives in (17) was proved in [26] to be rational in x_{1}, …, x_{n} and to have integral coefficients for any choice of p_{1}, …, p_{n}. Let f_{p_{1},... ,p_{n}} denote the denominator of the rational function (17) (we normalize the denominator so that the greatest common divisor of its coefficients equals 1). Our aim is to find the number of connected components of the amoeba complement ^{c}A_{f_{p_{1},... ,pn}}. For any fixed vector γ ∈ ℂ^{n}, Re γ_{i} ∈ [0, 1), there exist finitely many subsets of the shifted lattice ℤ^{n} +γ which satisfy the conditions in Proposition 3 and are contained in some strongly convex affine cone. We call them γ-admissible sets associated with (1). A set is said to be admissible if it is γ-admissible for some γ.

Proposition 9 The number of connected components of the amoeba complement cA_{f_{p_{1},... ,pn}} of the denominator of the Bergman kernel K_{p_{1},... ,p_{n}}(x) equals n + 1.

Remark 3 The conclusion of Proposition 9 can be deduced from Proposition 4.2 in [4] in the following way. Let us introduce new variables ξ_{i} = x_{i}^{1/p_{i}}. It follows from Proposition 4.2 in [4] that for any choice of the indices j_{1} ∈ {1, ..., p_{1}}, ..., j_{n} ∈ {1, ..., p_{n}} the amoeba of the first-order polynomial 1 − ε_{j_{1}1}ξ_{1} − ... − ε_{j_{n}n}ξ_{n} is the same. By Corollary 4.5 in [4] the number of connected components of its complement equals n + 1. Since a monomial change of the variables x_{1}, ..., x_{n} corresponds to a linear transformation of the amoeba space (see [4]), it follows that the number of the connected components of the complement of the amoeba of f_{p_{1},...,p_{n}} also equals n + 1. This shows in particular that the amoeba of f_{p_{1},...,p_{n}} is minimal.

We give here another proof of Proposition 9 which only uses hypergeometric properties of the Bergman kernels and does not use the explicit form of their denominators. Proof of Proposition 9 The Newton polytope of f_{p_{1},... ,p_{n}} has nonzero n-dimensional volume. Indeed, the restriction of K_{p_{1},... ,p_{n}}(x) to the complex line x_{1} = … [i] … = x_{n} = 0 is a rational function whose denominator is given by (1 − x_{i})^{k_{i}}, k_{i} > 0, i = 1, …, n. (Here [i] is the sign of omission.) It follows by Theorem B that the number of connected components of the amoeba complement ^{c}A_{f_{p_{1},... ,pn}} cannot be smaller than n + 1.

Let φ(s) denote the coefficient of the series (16), i.e.

φ(s) = \frac{Γ(p_{1}(s_{1} + 1) + … + p_{n}(s_{n} + 1) + 1)}{∏_{i=1}^{n} p_{i}Γ(p_{i}(s_{i} + 1))}.

Since for any i = 1, …, n the function φ(s) satisfies the equation

φ(s + e_{i}) ∏_{j=0}^{p_{i}−1}(p_{i}(s_{i} + 1) + j) = φ(s) ∏_{j=1}^{p_{i}}(p_{1}(s_{1} + 1) + … + p_{n}(s_{n} + 1) + j),

it follows that K_{p_{1},... ,p_{n}}(x) is a solution to the following system of the Horn type

x_{i} (∏_{j=1}^{p_{i}}(p_{1}(θ_{1} + 1) + … + p_{n}(θ_{n} + 1) + j)) K_{p_{1},... ,p_{n}}(x) =

(∏_{j=0}^{p_{i}−1}(p_{i}θ_{i} + j)) K_{p_{1},... ,p_{n}}(x), i = 1, …, n.  (18)

The number of irreducible 0-admissible sets associated with the system (18) equals n + 1. These sets are S_{0} = ℕ_{0}^{n} and S_{i} = {s ∈ ℤ^{n} : p_{1}(s_{1} + 1) + … + p_{n}(s_{n} + 1) + 1 ≤ 0, s_{j} ≥ 0, j ≠ i}, i = 1, …, n. (Notice that (16) is supported in the 0-admissible set ℕ_{0}^{n}.) Since any expansion of a rational solution to a Horn system into a Laurent series with the center at the origin is supported in an irreducible 0-admissible set, it follows that the number of connected components of the amoeba complement ^{c}A_{f_{p_{1},... ,pn}} cannot exceed n + 1. We have proved earlier that Newton polytope of f_{p_{1},... ,p_{n}} has at least n + 1 vertices. Thus it follows from Theorem B that the number of connected components of ^{c}A_{f_{p_{1},... ,pn}} cannot be smaller than n + 1 and hence equals n + 1. The proof is complete. □

Example 2 Let n = 2, p_{1} = 3, p_{2} = 2. The denominator of the Bergman kernel of the domain D^{3,2} is given by

f_{3,2}(x) = (1 − 2x_{1} − 3x_{2} + x_{1}^{2} − 6x_{1}x_{2} + 3x_{2}^{2} − x_{2}^{3})^{3}.

By Proposition 9 the number of connected components of the amoeba complement ^{c}A_{f_{3,2}} equals 3.

The Bergman kernel (16) gives an example of a rational hypergeometric function. The problem of describing the class of rational hypergeometric functions was studied in [2] and [3]. Observe however, that the definition of a hypergeometric function used in these papers is based on the Gelfand-Kapranov-Zelevinsky system of differential equations [7] rather than the Horn system.

6 Meromorphic nonconfluent hypergeometric functions are rational

The aim of this Section is to show that a nonconfluent Horn system (1) cannot possess a meromorphic solution different from a rational function (Theorem 11).

The relation between the support of a general Puiseux series and its domain of convergence is described by the Abel lemma (see Introduction and [8], § 1). For hypergeometric series the following stronger version of this statement holds.

Lemma 10 (Two-sided Abel lemma) Suppose that a nonconfluent hypergeometric Puiseux series y(x) with the support S has nonempty domain of convergence D. Let C be the cone of S. Then for any x^{(0)} ∈ D and for some x^{(1)} ∈ ℂ^{n} \ D

Log (x^{(0)}) − C^{∨} ⊂ Log (D) ⊂ Log (x^{(1)}) − C^{∨}.

Proof. The first inclusion follows from the general Abel lemma (see Introduction). Let us prove the second inclusion. Let M ⊂ ℝ^{n} be the lattice generated by the elements of the set S and let C denote the cone of S. By Proposition 5 the domain D is independent on the parameters c_{1}, …, c_{p} of the coefficient (4) (as long as they remain generic). Thus we may without loss of generality assume that S = C ∩ M. Since D is nonempty, it follows by Proposition 5 that C is a strongly convex polyhedral cone. Let u^{(1)}, …, u^{(N)} ∈ M denote the generators of C, i.e. C = {λ_{1}u^{(1)} + … + λ_{N}u^{(N)} : λ_{j} ≥ 0, j = 1, …, N}. For each j = 1, …, N we consider the restricted series y_{j}(x) = ∑_{k=0}^{∞} φ(ku^{(j)})x^{ku^{(j)}}. The nonconfluency condition (5) implies that ∑_{i=1}^{p}⟨A_{i}, u^{(j)}⟩ = 0. By the result on convergence of the generalized hypergeometric series in one variable (see [7], § 1.1) the domain of convergence of y_{j}(x) is contained in the set {x ∈ ℂ^{n} : |x^{u^{(j)}}| < r_{j}} for some constant r_{j} > 0. This shows that Log (D) ⊂ {v ∈ ℝ^{n} : ⟨u^{(j)}, v⟩ < log r_{j}, j = 1, …, N}. Since C is strongly convex, we can choose ξ ∈ ℝ^{n} such that m_{j} := ⟨u^{(j)}, ξ⟩ > 0. Define

x^{(1)} = Log^{−1} (ξ max_{j=1,... ,N} \frac{log r_{j}}{m_{j}} ) ,

then ⟨u^{(j)}, Log x^{(1)}⟩ ≥ log r_{j}, j = 1, …, N and hence Log (D) ⊂ {v ∈ ℝ^{n} : ⟨u^{(j)}, v − Log x^{(1)}⟩ < 0, j = 1, …, N} = Log x^{(1)} − C^{∨}. The proof is complete. □

The two-sided Abel lemma enables us to prove the following Theorem which is the main result in this Section.

Theorem 11 Any meromorphic nonconfluent hypergeometric function is rational.

Proof. Let y(x) be a meromorphic nonconfluent hypergeometric function. By definition y(x) is a solution to the Horn system (1). Since y(x) is nonconfluent, it follows by Proposition 5 and the two-sided Abel lemma (Lemma 10) that the domain of convergence of any shifted Laurent series representing y(x) is not all of (ℂ^{∗})^{n}. Therefore, using the assumption that y(x) is meromorphic, we can write it in the form h(x)/g(x), where h(x) is entire and g(x) is some polynomial which is not a monomial. This polynomial is given by the product of some irreducible factors in the resultant of (1) (see Section 2).

To prove that the function y(x) is rational it suffices to show that y(x) depends rationally on any given variable x_{i}, the other variables being fixed. Let us first consider the case when the Newton polytope N = N_{g} of the polynomial g(x) has zero n-dimensional volume (for examples of such rational hypergeometric functions see Example 1 in Section 3). Let T ⊂ ℝ^{n} denote the minimal linear subspace whose translation contains the polytope N . Choose a basis u_{1}, ..., u_{n} ∈ ℤ^{n} of the lattice ℤ^{n} such that u_{1}, ..., u_{m} is a basis of the sublattice T ∩ ℤ^{n}. Let us introduce new variables ξ_{i} = x^{u_{i}} = x_{1}^{u_{i1}}...x_{n}^{u_{in}}, i = 1, ..., n. We may without loss of generality assume that the determinant of the matrix U with the rows u_{1}, ..., u_{n} equals 1. It suffices to show that the function y(x(ξ)) depends rationally on the variables ξ_{1}, ..., ξ_{n}.

By the construction the polynomial g(ξ) is given by the product of a monomial and another polynomial which only depends on the variables ξ_{1}, ..., ξ_{m}. The Newton polytope of g(ξ) has nonzero m-dimensional volume. It follows by the two-sided Abel lemma that the cone of the support of any Laurent series ∑_{s∈ℤ^{n}} φ(s)ξ^{s} representing the function y(ξ) is contained in the linear subspace s_{m+1} = ... = s_{n} = 0. Hence y(ξ) depends polynomially on the variables ξ_{m+1}, ..., ξ_{n}. Let ξ = (ξ′, ξ′′), where ξ′ = (ξ_{1}, ..., ξ_{m}), ξ′′ = (ξ_{m+1}, ..., ξ_{n}). With these notations the function y(ξ) can be written in the form y(ξ) = ∑_{α∈W} a_{α}ξ′′^{α}y_{α}(ξ′), where W is a finite subset of the lattice ℤ^{n−m}, y_{α}(ξ′) is a meromorphic function depending on the variables ξ_{1}, ..., ξ_{m} only and a_{α} ∈ ℂ. We will prove that y_{α}(ξ′) is a hypergeometric function for any α ∈ W. Since the Newton polytope of g(ξ) has nonzero m-dimensional volume, this will show that it suffices to consider the case when the Newton polytope of the polynomial defining the singular set of a meromorphic hypergeometric function has the maximal possible dimension.

Let E_{i}^{λ_{i}} denote the operator which increases the ith argument of a function depending on n variables by λ_{i}, i.e. E_{i}^{λ_{i}}f(x) = f(x + λ_{i}e_{i}). For λ ∈ ℝ^{n} we denote the composition of the operators E_{1}^{λ_{1}}, ..., E_{n}^{λ_{n}} by E^{λ}, that is E^{λ}f(x) = f(x_{1} + λ_{1}, ..., x_{n} + λ_{n}). Since the commutator [θ_{i}, x_{j}^{λ_{j}}] equals δ_{ij}λ_{j}x_{j}^{λ_{j}}, it follows that for any polynomial P in n variables and any λ ∈ ℤ^{n}

P (θ)x^{λ} = x^{λ}(E^{λ}P)(θ).  (19)

By the definition the function y(x) is hypergeometric and hence satisfies the Horn system (1). Using the relation (19) and the ith equation of (1) we compute

x_{i}^{2}(E_{i}^{1}P_{i})(θ)P_{i}(θ)y(x) = (x_{i}P_{i}(θ))^{2}y(x) = x_{i}P_{i}(θ)Q_{i}(θ)y(x) =
(E_{i}^{−1}Q_{i})(θ)x_{i}P_{i}(θ)y(x) = (E_{i}^{−1}Q_{i})(θ)Q_{i}(θ)y(x).

Repeating this argument λ_{i} times we arrive at the formula

x_{i}^{λ_{i}} ( ∏_{j=0}^{λ_{i}−1}(E_{i}^{j}P_{i})(θ)) y(x) = ( ∏_{j=0}^{λ_{i}−1}(E_{i}^{−j}Q_{i})(θ)) y(x),  (20)

which holds for any λ_{i} ∈ ℕ. For u_{ki} ≥ 0 define polynomials ρ_{ki}(s) = ∏_{j=0}^{u_{ki}−1} E_{i}^{j}P_{i}(s) and τ_{ki}(s) = ∏_{j=0}^{u_{ki}−1} E_{i}^{−j}Q_{i}(s) (by the definition the empty product equals 1). For u_{ki} < 0 define polynomials ρ_{ki}(s) = ∏_{j=0}^{−u_{ki}−1} E_{i}^{−j}Q_{i}(s), τ_{ki}(s) = ∏_{j=0}^{−u_{ki}−1} E_{i}^{j}P_{i}(s). It follows from (20) that for any k = 1, ..., n

x_{i}^{u_{ki}}ρ_{ki}(θ)y(x) = τ_{ki}(θ)y(x), i = 1, ..., n.  (21)

Composing the operators in the equations (21) in the same way as we did before in order to obtain the formula (20), we arrive at the system of equations

x^{u_{k}} (∏_{j=1}^{n} ( ∏_{l=j+1}^{n} E_{l}^{u_{kl}}) ρ_{kj}(θ)) y(x) = (∏_{j=1}^{n} (∏_{l=1}^{j−1} E_{l}^{−u_{kl}}) τ_{kj}(θ)) y(x), k = 1, ..., n. (22)

For instance,

x_{1}^{u_{k1}}x_{2}^{u_{k2}}(E_{2}^{u_{k2}}ρ_{k1})(θ)ρ_{k2}(θ)y(x) = (by (19)) = x_{1}^{u_{k1}}ρ_{k1}(θ)x_{2}^{u_{k2}}ρ_{k2}(θ)y(x) =
(by the 2nd equation in (21)) = x_{1}^{u_{k1}}ρ_{k1}(θ)τ_{k2}(θ)y(x) = (by (19)) =
(E_{1}^{−u_{k1}}τ_{k2})(θ)x_{1}^{u_{k1}}ρ_{k1}(θ)y(x) = (by the 1st equation in (21)) =
(E_{1}^{−u_{k1}}τ_{k2})(θ)τ_{k1}(θ)y(x).

Each equation in (22) is obtained by repeating this argument n times.

Making in (22) the change of variables ξ_{i} = x^{u_{i}} and using the equality θ_{i} = x_{i} \frac{∂}{∂x_{i}} = u_{1i}ξ_{1} \frac{∂}{∂ξ_{1}} + ... + u_{ni}ξ_{n} \frac{∂}{∂ξ_{n}} , we conclude that y(ξ) is a solution to the system of equations

ξ_{i}ρ^{(i)}(θ_{ξ})y(ξ) = τ^{(i)}(θ_{ξ})y(ξ), i = 1, ..., n,  (23)

where θ_{ξ} = (ξ_{1} \frac{∂}{∂ξ_{1}} , ..., ξ_{n} \frac{∂}{∂ξ_{n}} ) , U is the matrix with the rows u_{1}, ..., u_{n} and

ρ^{(i)}(s) = ∏_{j=1}^{n} ( ∏_{l=j+1}^{n} E_{l}^{u_{kl}}) ρ_{kj}((U^{T})^{−1}s),

τ^{(i)}(s) = ∏_{j=1}^{n} (∏_{l=1}^{j−1} E_{l}^{−u_{kl}}) τ_{kj}((U^{T})^{−1}s).

Since y(ξ) = ∑_{α∈W} a_{α}ξ′′^{α}y_{α}(ξ′), it follows from the first m equations of the system (23) that

(ξ_{i}ρ^{(i)}(θ_{ξ}) − τ^{(i)}(θ_{ξ})) y(ξ) = ∑_{α∈W} a_{α}ξ′′^{α} ((ξ_{i}ρ^{(i)}(θ_{ξ}) − τ^{(i)}(θ_{ξ})) y_{α}(ξ′)) = 0

for i = 1, ..., m. Since y_{α}(ξ′) does not depend on ξ_{m+1}, ..., ξ_{n}, it follows that for any α ∈ W

ξ_{i}ρ^{(i)}(θξ^{′})y_{α}(ξ′) = τ^{(i)}(θξ^{′})y_{α}(ξ′), i = 1, ..., m.  (24)

Here θξ^{′} = (ξ_{1} \frac{∂}{∂ξ_{1}} , ..., ξ_{m} \frac{∂}{∂ξ_{m}} , 0, ..., 0) . The system (24) is a Horn system in m variables. Thus the function y_{α}(x) is hypergeometric for any α ∈ W. By the assumption the Newton polytope of the polynomial defining the singularity of the meromorphic function y_{α}(ξ′) has dimension m. To prove that the original function y(x) is rational it suffices to show that y_{α}(ξ′) depends rationally on ξ_{1}, ..., ξ_{m} for any α ∈ W. Thus it is sufficient to prove the theorem in the case when the Newton polytope of the polynomial which defines the singular set of a given meromorphic hypergeometric function has the maximal possible dimension.

Suppose now that dim N = n. Let C_{v}^{∨} be the cone which is dual to N at the point v. By the remark after Theorem B to each vertex v of the polytope N one can associate a connected component of the amoeba complement ^{c}A_{g}. This component is the image of the domain of convergence of some Laurent series L_{v} for the function y(x) = h(x)/g(x) under the mapping Log. It contains some translation w_{v} + C_{v}^{∨} of the cone C_{v}^{∨}. By the two-sided Abel lemma the cone of the support of the series L_{v} coincides with the cone −(C_{v}^{∨})^{∨} = −C_{v}. The family of the cones {C_{v}^{∨}}_{v∈vert(N)} coincides with the set of all maximal cones of the dual fan Σ_{N} of the polytope N . Since for any polytope its dual fan is complete, it follows that the toric variety 𝕏_{Σ_{N}} associated with the fan Σ_{N} is compact (see § 2.4 in [6]). This variety can be covered by the affine toric varieties {U_{C_{v}^{∨}}}_{v∈vert(N)}.

It is known that the monomials {z^{α} : α ∈ −C_{v}} are holomorphic in U_{C_{v}^{∨}} (see § 1.3 in [6]). Since the cone of the support of the series L_{v} coincides with −C_{v}, it follows that for some w_{v} ∈ ℤ^{n} the series x^{w_{v}}L_{v} contains only those monomials which are holomorphic in U_{C_{v}^{∨}}. Thus x^{w_{v}}y(x) is holomorphic in U_{C_{v}^{∨}} for all v ∈ vert(N). This shows that the restriction of y(x) to any line x_{j} = const has polynomial growth in ℂ^{∗} and hence is rational. It is well-known that a function which is rational in each variable depends rationally on all of the variables. This completes the proof. □

Thanks to Theorem 11 we do not need to make any difference between meromorphic and rational nonconfluent hypergeometric functions. From now on we formulate all the results using the term “rational”.

Remark 4 Let f(x) be a rational function in n variables with singularities along an algebraic hypersurface V ⊂ ℂ^{n} and let A be the image of V under the mapping Log. By Theorem A the connected components of the amoeba complement ^{c}A are in bijective correspondence with the Laurent series expansions (with the center at the origin) of f(x). For a multi-valued analytic function F (x) with singularities on the same variety V this correspondence is in general not one-to-one. It may happen that some of the connected components of ^{c}A do not correspond to any expansion of F (x) since there is no holomorphic branch of F (x) on the pull-back of this component. It is also possible that several connected components of ^{c}A correspond to a single series expansion of F (x). (For instance, let x ∈ ℂ and consider the function F (x) = \sqrt{\sqrt{x + 2} + \sqrt{3}}. There exists a holomorphic branch of F (x) in the disk {|x| < 2} although x = 1 is a branching point. A similar situation in the two-dimensional case is described in Example 3.) However, with each series expansion of F (x) centered at the origin one can associate at least one connected component of ^{c}A.

7 Rational solutions to the Horn system

Typically a hypergeometric function is a multi-valued analytic function with singularities along an algebraic hypersurface (see Section 2). In this Section we give a necessary condition for a hypergeometric series to represent a germ of a rational function. This allows one to give an explicit description of the class of rational solutions to (1) in the case when for any i = 1, …, n the polynomial Q_{i}(s) depends on s_{i} only, each linear factor of P_{i}(s) depends on all the variables and the resultant of (1) is irreducible. We prove that any such rational hypergeometric function is contiguous to the Bergman kernel K_{p_{1},... ,p_{n}} for some p_{1}, …, p_{n} (Proposition 14).

Recall that B_{1}, ..., B_{d} are defined to be the duals to the maximal elements (with respect to inclusion) of the finite family {−C_{I}} of strongly convex polyhedral cones. Here C_{I} is the recession cone of the convex hull of the support of the hypergeometric series (13). Let X_{1}, ..., X_{N} denote the recession cones of the connected components of the amoeba complement^{c}A_{R(x)} of the resultant of (1). These recession cones are welldefined since by Theorem A the connected components of the amoeba complement are convex. The following Theorem describes the structure of the amoeba A_{R(x)}.

Theorem 12 Suppose that a nonconfluent Horn system possesses a rational solution with the poles on the zero set of its resultant R(x). Then the fan of this Horn system is well-defined and dual to the Newton polytope of R(x).

Proof. Since there exists a rational solution to (1) with the poles on the zero set of its resultant R(x) it follows by Theorems B and 7 that the cone X_{i} has nonempty interior for any i = 1, ..., N. Thus by Theorem B the cones {X_{i}}_{i=1}^{N} can be identified with the maximal cones of the fan which is dual to the Newton polytope of R(x).

It suffices to show that the family {B_{i}}_{i=1}^{d} consists of the same elements as the family {X_{i}}_{i=1}^{N}. As we have already seen in Section 4 the nonconfluency condition (5) for the Horn system (1) implies that ⋃_{j=1}^{d} B_{j} = ℝ^{n}. Hence for any i = 1, ..., N there exists k_{i} ∈ {1, ..., d} such that int(X_{i} ∩ B_{k_{i}}) ≠ ∅. Let L_{i} denote a series solution to (1) whose support S_{i} defines the cone B_{k_{i}} in the sense that B_{k_{i}} = −C_{S_{i}}^{∨}. Here C_{S_{i}} is the cone of S_{i} (see Section 4). Let L̃_{i} denote the series expansion of the rational solution to (1) such that the recession cone of the image of its domain of convergence under the mapping Log is X_{i}. Since int(X_{i} ∩B_{k_{i}}) ≠ ∅ it follows that the series L+L̃_{i} has a nonempty domain of convergence Ω_{i}. By the two-sided Abel lemma the cone of the convex set Log Ω_{i} is X_{i} ∩ B_{k_{i}}.

Any Puiseux series solution to (1) whose domain of convergence lies entirely in the pre-image of a connected component of the amoeba complement ^{c}A_{R(x)} with respect to the mapping Log converges on the whole of this pre-image. Using the two-sided Abel lemma we conclude that B_{k_{i}} cannot be a proper subset of X_{i}. Thus either X_{i} = B_{k_{i}} or B_{k_{i}}^{∨} is a proper subset of (X_{i} ∩ B_{k_{i}})^{∨}. The latter is impossible due to the assumption that B_{k_{i}}^{∨} is a maximal element in the family of the cones of the supports of series solutions to (1). Hence X_{i} = B_{k_{i}} for any i = 1, ..., N. Since the cones {X_{i}}_{i=1}^{N} are the maximal cones of a complete fan, it follows that d = N and thus we can identify the families of the cones {X_{i}}_{i=1}^{N} and {B_{i}}_{i=1}^{d}. The proof is complete. □

The conditions in Theorem 12 are sufficient for the fan of a Horn system to be dual to the Newton polytope of its resultant, but they are not necessary. For instance, the fan of the system (27) in Example 3 below is dual to the Newton polytope of its resultant though the system (27) has no nonzero rational solutions. Yet, the remark directly before Example 1 shows that the conclusion of Theorem 12 does not hold in arbitrary case.

Corollary 13 If a Horn system possesses a rational solution with the poles on the zero set of its resultant then the number of 0-admissible sets associated with this system cannot be smaller than the number of the maximal cones in its fan.

Proof. By Theorem 12 the fan of the Horn system is well-defined. Let y(x) be a rational solution to (1) with the poles on the zero set of the resultant R(x) of (1). By Theorem A the number of Laurent series expansions of y(x) with the center at the origin equals the number of connected components of the set ^{c}A_{R}. By Theorem 7 R(x) is a minimal polynomial and hence by Theorem 12 there exists a one-to-one correspondence between the connected components of ^{c}A_{R(x)} and the maximal cones of the fan of the system (1). Since any expansion of y(x) is supported in a 0-admissible set it follows by that the number of such sets cannot be smaller than the number of the maximal cones in the fan of the Horn system. This completes the proof of the Corollary. □

As we have seen in Section 2 a solution to the Horn system (1) can only be singular on the set on which the resultant R(x) of (1) vanishes. Typically R(x) is divisible by some monomial x^{a}, a ∈ ℕ_{0}^{n}. We denote the quotient R(x)/x^{a} (with the maximal possible |a| = a_{1} + ... + a_{n}) by r(x) and call it the essential resultant of the system (1). The reason for introducing this terminology is the fact that a Laurent monomial has unique Laurent series development with the center at the origin. Therefore such a monomial is an unessential factor as long as one is concerned with the problem of computing the number of connected components of the amoeba complement of a mapping.

We can say much more in the particular case when the polynomial Q_{i}(s) depends only on s_{i} for all i = 1, …, n. Under this assumption it is possible to compute the dimension of the space of holomorphic solutions to the Horn system (1) explicitly and construct a basis in this space if the parameters of the system are sufficiently general [19]. (Theorem 9 in [19] assumes that deg Q_{i} > deg P_{i}, i = 1, …, n, which is not the case if the nonconfluency relation (5) holds. Yet, by the Lemma in § 1.4 of [7] each of the basis series which were constructed in § 3 of [19] converges in some neighborhood of the origin if the original Horn system is nonconfluent. The multivalued analytic functions determined by these series give a global basis in the space of holomorphic solutions to (1).) Recall that two Ore-Sato coefficients (and the corresponding hypergeometric series) are called contiguous if their quotient can be reduced to the product of a rational function and an exponential term t̃_{1}^{s_{1}} … t̃_{n}^{s_{n}}. The next Proposition provides an explicit description of the class of rational solutions to such systems of hypergeometric type under some additional assumptions which are always satisfied if the parameters of the system are sufficiently general.

Proposition 14 Suppose that the nonconfluent Ore-Sato coefficient

ψ(s) = t_{1}^{s_{1}} … t_{n}^{s_{s}} \frac{∏_{i=1}^{p} Γ(⟨A_{i}, s⟩ − c_{i})}{∏_{j=1}^{n} ∏_{k=1}^{p_{j}} Γ(s_{j} − α_{jk})}  (25)

defines the Horn system (1) with the irreducible essential resultant r(x) and satisfies the conditions A_{ij} > 0, i = 1, …, p, j = 1, …, n. Let y(x) = ∑_{s∈ℕ_{0}^{n}} ψ(s)x^{s} and let A be the matrix with the rows A_{1}, …, A_{p}. If rank A > 1 then the series y(x) cannot define a rational function. (We disregard exceptional values of the parameters of ψ(s) for which y(x) reduces to a linear combination of hypergeometric series in fewer variables.) If rank A = 1 and y(x) is rational then it is contiguous to the series (16) converging to the Bergman kernel K_{p_{1},... ,p_{n}}(x).

Proof. Suppose that rank A > 1 and y(x) is a rational function. Without loss of generality we assume that A_{11}A_{22} − A_{12}A_{21} ≠ 0. For each m = 1, …, p we define the vector μ_{m} = (μ_{m1}, ..., μ_{mn}) ∈ ℕ^{n} by μ_{m} = ∑_{i=1}^{m} A_{i} and consider the nonconfluent Ore-Sato coefficient

χ_{m}(s) = \frac{∏_{i=1}^{m} Γ(⟨A_{i}, s⟩ − c_{i})}{∏_{j=1}^{n} ∏_{k=1}^{μ_{mj}} Γ(s_{j} − α_{jk})}.

Each of these coefficients defines a nonconfluent system of differential equations of the Horn type (see Remark 1). Let B_{m1}, …, B_{md_{m}} be the maximal elements in the family of the cones of the admissible sets associated with the system defined by χ_{m}(s) (see Section 4). Arguing as in the proof of Proposition 9 we conclude that d_{1} = n+1. Let Ã be the matrix with the rows A_{1}, A_{2}, e_{3}, …, e_{n}, c̃ = (c_{1}, c_{2}, 0, …, 0) ∈ ℂ^{n} and define γ to be the solution to the system of linear equations Ãs = c̃. The set {s ∈ ℤ^{n} + γ : Ãs ≥ 0} satisfies the conditions in Proposition 3 if the parameters c_{1}, …, c_{p} are generic. This yields d_{2} ≥ n + 2. By Remark 2 d_{i} ≤ d_{j} for i ≤ j. Since χ_{p}(s) = ψ(s) it follows by Theorem 12 that the number of connected components of the amoeba complement^{c}A_{r(x)} at least equals n+2. By the assumption the series y(x) represents a germ of a rational function. Since r(x) is irreducible, the function y(x) must be singular on the whole of the hypersurface {r(x) = 0}. Thus it follows from Theorem A that the number of Laurent series developments (centered at the origin) of this rational function at least equals n + 2. Yet, the condition A_{ij} > 0 and the conditions (8),(9) in Proposition 3 imply that the number of 0-admissible subsets associated with the Horn system defined by the Ore-Sato coefficient ψ(s) cannot exceed n + 1. This contradicts the conclusion of Corollary 13 and shows that the function y(x) cannot be rational unless rank A = 1.

Suppose now that rank A = 1 and that the series y(x) converges to a rational function. Let δ = GCD (p_{1}, …, p_{n}), p̃_{i} = p_{i}/δ, i = 1, …, n. It follows from the nonconfluency condition ∑_{i=1}^{p} A_{i} = (p_{1}, …, p_{n}) and the Gauss multiplication formula for the Γ-function that ψ(s) is contiguous to ψ̃(s) = ∏_{l=0}^{δ−1} Γ(p̃_{1}s_{1} + … + p̃_{n}s_{n} + a_{l})/ ∏_{j=1}^{n} ∏_{k=1}^{p_{j}} Γ(s_{j} − α_{jk}). Here a_{0}, …, a_{δ−1} ∈ ℂ are some constants. Moreover the quotient ψ(s)/ψ̃(s) is given by an exponential term t̃_{1}^{s_{1}} … t̃_{n}^{s_{n}} and hence the series ỹ(x) = ∑_{s∈ℕ_{0}^{n}} ψ̃(s)x^{s} converges to a rational function. By the assumption p̃_{i} ≠ 0 for any i = 1, ..., n. The restriction of ỹ(x) to the complex line x_{1} = ...[i]... = x_{n} = 0 is a rational function (here [i] is the sign of omission). Let ψ̃_{i}(s_{i}) = ψ̃(0, ..., s_{i}, ..., 0) (s_{i} in the ith position). Using once again the Gauss multiplication formula we conclude that the series

∑_{s_{i}=0}^{∞} \frac{∏_{l=0}^{δ−1} ∏_{j=0}^{p̃_{i}−1} Γ (s_{i} + \frac{a_{l}+j}{p̃_{i}} )}{∏_{k=0}^{p_{i}−1} Γ (s_{i} + \frac{k}{p_{i}} )}x_{i}^{s_{i}}

represents a rational function. A criterion for a power series in one variable to converge to a rational function (see Theorem 4.1.1 in [23]) implies that for any l = 0, ..., δ − 1, j = 0, ..., p̃_{i} − 1 there exists k ∈ {0, ..., p_{i} − 1} such that (a_{l} + j)/p̃_{i} − k/p_{i} ∈ ℕ. Hence for any l = 0, ..., δ − 1 one can find k ∈ {0, ..., p_{i} − 1} such that a_{l} − k/δ ∈ ℤ. Thus ψ(s) is contiguous to the Ore-Sato coefficient

\frac{∏_{l=0}^{δ−1} Γ (p̃_{1}s_{1} + … + p̃_{n}s_{n} + \frac{l}{δ})}{∏_{j=1}^{n} ∏_{k=0}^{p_{j}−1} Γ(s_{j} + \frac{k}{p_{j}} + 1)}.

The Gauss multiplication formula shows that the latter coefficient is contiguous to the coefficient of the series (16) which represents the Bergman kernel K_{p_{1},... ,p_{n}}. The proof is complete. □

Let us now consider an example. This example deals with a simplified version of the hypergeometric series which expresses a solution y(x) to the cubic equation y^{3} + x_{1}y^{2} + x_{2}y − 1 = 0 in terms of the coefficients x_{1}, x_{2} (see [14],[21],[24] and Corollary 8).

Example 3 Consider the hypergeometric series

y(x_{1}, x_{2}) = ∑_{s_{1},s_{2}≥0} \frac{Γ(2s_{1} + s_{2} + α)Γ(s_{1} + 2s_{2} + β)}{Γ(3s_{1} + 3)Γ(3s_{2} + 3)}x_{1}^{s_{1}}x_{2}^{s_{2}},  (26)

where α, β are arbitrary parameters such that the coefficient of the series is welldefined and different from zero on ℕ_{0}^{2}. By the Lemma in § 1.4 of [7] the series (26) converges in some neighborhood of the origin. This series satisfies the system of equations of hypergeometric type

\{ \begin{matrix}x_{1}(2θ_{1} + θ_{2} + α)(2θ_{1} + θ_{2} + α + 1)(θ_{1} + 2θ_{2} + β)y(x) = \\ 3θ_{1}(3θ_{1} + 1)(3θ_{1} + 2)y(x), \\ x_{2}(2θ_{1} + θ_{2} + α)(θ_{1} + 2θ_{2} + β)(θ_{1} + 2θ_{2} + β + 1)y(x) = \\ 3θ_{2}(3θ_{2} + 1)(3θ_{2} + 2)y(x). \end{matrix}  (27)

The principal symbols of the operators in (27) are

\begin{matrix}H_{1}(x, z) = & x_{1}(2x_{1}z_{1} + x_{2}z_{2})^{2}(x_{1}z_{1} + 2x_{2}z_{2}) − 27(x_{1}z_{1})^{3}, \\ H_{2}(x, z) = & x_{2}(2x_{1}z_{1} + x_{2}z_{2})(x_{1}z_{1} + 2x_{2}z_{2})^{2} − 27(x_{2}z_{2})^{3}. \end{matrix}

The singular locus of a solution to (27) is contained in the set on which the polynomials H_{1}(x, z), H_{2}(x, z) (considered as polynomials in z_{1}, z_{2} whose coefficients depend on the parameters x_{1}, x_{2}) do not form a regular sequence (see the remark after the proof of Proposition 2). This happens if and only if the resultant of H_{1}(x, z), H_{2}(x, z) with respect to z_{1}, z_{2} is equal to zero. This resultant is given by

R(x_{1}, x_{2}) = x_{1}^{9}x_{2}^{9}(x_{1}^{2}x_{2}^{2} + 64x_{1}^{3} − 24x_{1}^{2}x_{2} − 24x_{1}x_{2}^{2} + 64x_{2}^{3}

−1296x_{1}^{2} + 4698x_{1}x_{2} − 1296x_{2}^{2} + 8748x_{1} + 8748x_{2} − 19683).  (28)

The essential resultant r(x_{1}, x_{2}) = R(x_{1}, x_{2})/(x_{1}x_{2})^{9} of the system (27) is an irreducible polynomial. The vectors (2, 1), (1, 2) of the coefficients of the linear factors in the arguments of the Γ-functions in the numerator of the coefficient of (26) are linearly independent. By Proposition 14 the series (26) cannot converge to a rational function.

[figure: .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .; .]

Fig. 4 The maximal cones of the irreducible supports of solutions to (27)

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Fig. 5 The amoeba of the essential resultant of the Horn system (27)

[figure]

Fig. 6 The Newton polytope of the essential resultant of the system (27)

The fact that the series (26) cannot define a germ of a rational function can be seen without appealing to Proposition 14 since we have the explicit expression (28) for the resultant of the principal symbols of the differential operators in (27). If the sum y(x) of the series (26) was rational then by Theorem 7 the number of expansions of y(x) into a Laurent series with the center at the origin would be equal to 4 since the Newton polytope of the essential resultant of (27) has 4 vertices (see Figure 6). However, Proposition 3 shows that for any choice of the parameters α, β at most 3 of the admissible subsets can belong to ℤ^{2} (see Figure 4). Thus the sum of the series (26) is not a rational function.

To determine the resultant of a general Horn system is a problem of great computational complexity. Theorem 12 and the corollary to it allow one to describe the amoeba of the resultant of a Horn system and draw consequences on its solvability in the class of rational functions without performing this computation.

Remark 5 There exist rational hypergeometric functions which cannot be described in terms of the Bergman kernels of complex ellipsoidal domains. For instance, the hypergeometric series

∑_{s∈ℕ_{0}^{n}} \frac{Γ(s_{1} + p(s_{2} + … + s_{n} + 1))Γ(s_{2} + … + s_{n} + 1)}{Γ(s_{1} + 1) … Γ(s_{n} + 1)Γ(p(s_{2} + … + s_{n} + 1))}x^{s} =

((1 − x_{1})^{p} − x_{2} − … − x_{n})^{−1}

is not contiguous to such a kernel whenever n ≥ 3, p ≥ 2.

References

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DEPARTMENT OF MATHEMATICS UNIVERSITY OF STOCKHOLM S-10691 STOCKHOLM SWEDEN

E-mail: passare@matematik.su.se timur@matematik.su.se

DEPARTMENT OF MATHEMATICS KRASNOYARSK STATE UNIVERSITY 660041 KRASNOYARSK RUSSIA

E-mail: tsikh@lan.krasu.ru

and

MAX-PLANCK-INSTITUT

FÜR MATHEMATIK

D-53111 BONN GERMANY

E-mail: tsikh@mpim-bonn.mpg.de