Sparse hypergeometric systems Текст научной статьи по специальности «Математика»

y^K 517.55

SPARSE HYPERGEOMETRIC SYSTEMS Timur Sadykov

Institute of Mathematics, Siberian Federal University, pr. Svobodny, 79, Krasnoyarsk, 660041, Russia, e-mail: [email protected]

Abstract. We study the approach to the theory of hypergeometric functions in several variables via a generalization of the Horn system of differential equations. A formula for the dimension of its solution space is given. Using this formula we construct an explicit basis in the space of holomorphic solutions to the generalized Horn system under some assumptions on its parameters.

Keywords: hypergeometric functions, Horn system of differential equations, Mellin system.

1 Introduction

There exist several approaches to the notion of a hypergeometric function depending on several complex variables. It can be defined as the sum of a power series of a certain form (such series are known as T-series) [10], as a solution to a system of partial differential equations [9], [11], [1], or as a Mellin-Barnes integral [15]. In the present paper we study the approach to the theory of hypergeometric functions via a generalization of the Horn system of differential equations. We consider the system of partial differential equations of hypergeometric type

xUiPiip)y(x) = Qi(9)y(x), i =1 (1.1)

where the vectors ui = (uii,... ,uin) G Zn are assumed to be linearly independent, Pi,Qi are nonzero polynomials in n complex variables and 6 = {6i, ... , 6n), Q% = Xi^. We use the notation Xj ' 'X ^ ... Xn' . If {ui}n=1 form the standard basis of the lattice Zn then the system (1.1) coincides with a classical system of partial differential equations which goes back to Horn and Mellin (see [13] and § 1.2 of [10]). In the present paper the system (1.1) is referred to as the sparse hypergeometric system (or generalized Horn system) since, in general, its series solutions might have many gaps.

A sparse hypergeometric system can be easily reduced to the classical Horn system by a monomial change of variables. The main purpose of the present paper is to discuss the relation between the sparse and the classical case in detail for the benefit of a reader interested in explicit solutions of hypergeometric D-modules. We also furnish several examples which illustrate crucial properties of the singularities of multivariate hypergeometric functions. Most of the statements in this article are parallel to or follow from the results in [16].

A typical example of a sparse hypergeometric system is the Mellin system of equations (see [7]). One of the reasons for studying sparse hypergeometric systems is the fact that knowing the structure of solutions to (1.1) allows one to investigate the so-called amoeba of the singular locus of a solution to (1.1). The notion of amoebas was introduced by Gelfand, Kapranov and Zelevinsky (see [12], Chapter 6, § 1). Given a mapping f (x), its amoeba Af is the image of the hypersurface f-1(0) under the map (x1,..., xn) ^ (log |x1|,...,log |xn|). In section 5 we use the

The author was supported by the Russian Foundation for Basic Research, grant 09-01-00762-a, by grant no. 26 for scientific research groups of Siberian Federal University and by the "Dynasty"foundation.

results on the structure of solutions to (1.1) for computing the number of connected components of the complement of amoebas of some rational functions. The problem of describing the class of rational hypergeometric functions was studied in a different setting in [5], [6]. The definition of a hypergeometric function used in these papers is based on the Gelfand-Kapranov-Zelevinsky system of differential equations [9], [10], [11].

Solutions to (1.1) are closely related to the notion of a generalized Horn series which is defined as a formal (Laurent) series

y(x) = (1.2)

seZ"

whose coefficients <^(s) are characterized by the property that p(s + щ) = ip(s)Ri(s). Here Ri(s) are rational functions. We also use notations y = (71,..., jn) G Cn, Re ji G [0,1), xs = x! .. .хП. In the case when {ui}'^=1 form the standard basis of Zn we get the definition of the classical Horn series (see [10], § 1.2).

In the case of two or more variables the generalized Horn system (1.1) is in general not solvable in the class of series (1.2) without additional assumptions on the polynomials Pi,Qi. In section 2 we investigate solvability of hypergeometric systems of equations and describe supports of solutions to the generalized Horn system. The necessary and sufficient conditions for a formal solution to the system (1.1) in the class (1.2) to exist are given in Theorem 2.1.

In section 3 we consider the D-module associated with the generalized Horn system. We give a formula which allows one to compute the dimension of the space of holomorphic solutions to (1.1) at a generic point under some additional assumptions on the system under study (Theorem 3.3). We give also an estimate for the dimension of the solution space of (1.1) under less restrictive assumptions on the parameters of the system (Corollary 3.4).

In section 4 we consider the case when the polynomials Pi,Qi can be factorized up to polynomials of degree 1 and construct an explicit basis in the space of holomorphic solutions to some systems of the Horn type. We show that in the case when Ri(s+Uj)Rj(s) = Rj (s+ui)Ri(s), Qi(s + Uj) = Qi(s) and degQi(s) > degPi(s), i, j = 1,... ,n, i = j, there exists a basis in the space of holomorphic solutions to (1.1) consisting of series (1.2) if the parameters of the system under study are sufficiently general (Theorem 4.1).

In section 5 we apply the results on the generalized Horn system to the problem of describing the complement of the amoeba of a rational function. We show how Theorem 2.1 can be used for studying Laurent series developments of a rational solution to (1.1). A class of rational hypergeometric functions with minimal number of connected components of the complement of the amoeba is described.

2 Supports of solutions to sparse hypergeometric systems

Suppose that the series (1.2) represents a solution to the system (1.1). Computing the action of the operator xUiPi(9) — Qi(9) on this series we arrive at the following system of difference equations

<^(s + Ui)Qi(s + y + Ui) = <p(s)Pi(s + y), i = 1,..., n. (2.1)

The system (2.1) is equivalent to (1.1) as long as we are concerned with those solutions to the generalized Horn system which admit a series expansion of the form (1.2). Let Zn + y denote the shift in Cn of the lattice Zn with respect to the vector y. Without loss of generality we assume

that the polynomials Pi(s), Qi(s + ui) are relatively prime for all i = 1,..., n. In this section we shall describe nontrivial solutions to the system (2.1) (i.e. those ones which are not equal to zero identically). While looking for a solution to (2.1) which is different from zero on some subset S of Zn we shall assume that the polynomials Pi(s),Qi(s), the set S and the vector y satisfy the condition

|pi(s + Y)| + |Qi(s + Y + ui)| = ° (2.2)

for any s G S and for all i = 1,..., n. That is, for any s G S the equality Pi(s + y) =0 implies that Qi(s + y + ui) = 0 and Qi(s + y + ui) = 0 implies Pi(s + y) = 0.

The system of difference equations (2.1) is in general not solvable without further restrictions on Pi, Qi. Let Ri(s) denote the rational function Pi(s)/Qi(s + ui), i = 1,..., n. Increasing the argument s in the ith equation of (2.1) by uj and multiplying the obtained equality by the jth equation of (2.1), we arrive at the relation <^(s + ui + uj)/<^(s) = Ri(s + uj)Rj(s). Analogously, increasing the argument in the jth equation of (2.1) by ui and multiplying the result by the ith equation of (2.1), we arrive at the equality <^(s + ui + uj)/<^(s) = Rj(s + ui)Ri(s). Thus the conditions

Ri(s + uj)Rj(s) = Rj(s + ui)Ri(s), i,j =1,...,n (2.3)

are in general necessary for (2.1) to be solvable. The conditions (2.3) will be referred to as the compatibility conditions for the system (2.1). Throughout this paper we assume that the polynomials Pi,Qi defining the generalized Horn system (1.1) satisfy (2.3).

Let U denote the matrix whose rows are the vectors u1,... ,un. A set S C Zn is said to be U-connected if any two points in S can be connected by a polygonal line with the vectors u1,... ,un as sides and vertices in S. Let <^(s) be a solution to (2.1). We define the support of <^(s) to be the subset of the lattice Zn where <^(s) is different from zero. A formal series x7 ^(s)xs is called a formal solution to the system (1.1) if the function <^(s) satisfies

the equations (2.1) at each point of the lattice Zn. The following Theorem gives necessary and sufficient conditions for a solution to the system (2.1) supported in some set S C Zn to exist.

Theorem 2.1 For S C Zn define

Si = {s G S : s + ui G S}, Si = {s G S : s + ui G S}, i = 1,. .., n.

Suppose that the conditions (2.2) are satisfied on S. Then there exists a solution to the system

(2.1) supported in S if and only if the following conditions are fulfilled:

Pi(s + Y)Is = 0, Qi(s + y + ui)|S" = 0, i = 1,..., n, (2.4)

ii

Pi(s + Y)IS\S = 0, Qi(s + Y + ui)|s = 0, i = 1,..., n. (2.5)

The proof of this theorem is analogous to the proof of Theorem 1.3 in [16]. Theorem 2.1 will be used in section 4 for constructing an explicit basis in the space of holomorphic solutions to the generalized Horn system in the case when degQi > degPi and Qi(s + uj) = Qi(s),

i, j = 1,. .., n, i = j. In the next section we compute the dimension of the space of holomorphic solutions to (1.1) at a generic point.

3 Holomorphic solutions to sparse systems

Let Gi denote the differential operator xuiPi(0) — Qi(0), i = 1,..., n. Let D be the Weyl algebra

in n variables [3], and define M = D/^n=1 DGi to be the left D-module associated with the

system (1.1). Let R = C[z1,..., zn] and R[x] = R[x1,..., xn] = C[x1,..., xn, z1,..., zn]. We

make R[x] into a left D-module by defining the action of cj on R[x] by

_d_

dx,

+ z,.

Let $ : D ^ R[x] be the D-linear map defined by

Ф(х?

ai

■aï1... an- )

x“i Jb ï

rpa— y b1

..xn z1

(3.1)

(3.2)

It is easily checked that $ is an isomorphism of D-modules. In this section we establish some properties of linear operators acting on R[x]. We aim to construct a commutative family of D-linear operators Wi : R[x] ^ R[x], i = 1,..., n which satisfy the equality $(Gi) = Wi(1). The crucial point which requires additional assumptions on the parameters of the system (1.1) is the commutativity of the family {Wi}n=1 which is needed for computing the dimension (as a C-vector space) of the module R[x]/J^’=1 WiR[x] at a fixed point x(0). We construct the operators Wi and show that they commute with one another under some additional assumptions on the polynomials Qi(s) (Lemma 3.1). However, no additional assumptions on the polynomials Pi(s) are needed as long as the compatibility conditions (2.3) are fulfilled.

Following the spirit of Adolphson [1] we define operators Di : R[x] ^ R[x] by setting

, n.

(3.3)

It was pointed out in [1] that the operators (3.3) form a commutative family of D-linear operators. Let D denote the vector (D1,...,Dn). For any i = 1,...,n we define operator Vi : R[x] ^ R[x] by Vi = zi-1Di. This operator commutes with the operators cj since both Di and the multiplication by z-1 commute with cj. Moreover, the operator Vi commutes with Vj for all 1 < i, j < n and with Dj for i = j. In the case i = j we have ViDi = Vi + DiVi.

Thanks to Lemma 2.2 in [16] we may define operators Wi = Pi(D)VUi — Qi(D) such that for any i = 1,...,n Wi is a D-linear operator satisfying the identity $(Gi) = Wi(1). It follows by the D-linearity of Wi that J^n=1 WiR[x] and R[x]^J^n=1 WiR[x] can be considered as left D-modules. Using Theorem 4.4 and Lemma 4.12 in [1], we conclude that the following isomorphism holds true:

(3.4)

In the general case the operators W = Pi(D)VUi — Qi(D) do not commute since does not commute with Vj. However, this family of operators may be shown to be commutative under some assumptions on the polynomials Qj(s) in the case when the polynomials Pj(s),Qj(s) satisfy the compatibility conditions (2.3). The following Lemma holds.

b

n

z

n

Lemma 3.1 The operators Wj = Pj(D)VUi — Qj(D) commute with one another if and only if the polynomials Pj(s),Qj(s) satisfy the compatibility conditions (2.3) and for any = 1,...,n, i = j, Qj(s + uj) = Qj(s).

Proof Since Vi = zi + Dizi it follows that ViDi = Vi + DiVi and that Vi commutes

with Dj for i = j. Hence for any a = (a1,... , an) G Nn

ViDa1 ... Dnn = Da1... (Di + 1)ai... Dnn Vi. (3.5)

Let E* denote the operator which increases the ith argument by t, that is, E*f(x) = / (x + tei).

n

i=1

Here {ei}n=1 denotes the standard basis of Zn. It follows from (3.5) that

ViPj (D) = (E1Pj )(D)Vi. (3.6)

For a G Zn let Ea denote the composition Ea1 o ... o E^". Using (3.6) we compute the

commutator of the operators Wi, Wj :

WiWj — Wj Wi = ^Pi(D)(EUi Pj )(D) — Pj (D)(EUj Pi)(D)) Vui+u +

((EUj Qi)(D) — Qi(D)) Pj (D)VUj + (Qj (D) — (EUi Qj )(D)) Pi(D)VUi. (3.7)

Let us define the grade g(xaz^) of an element xaz^ of the ring R[x] to be a — ft. Notice that g(Di(xaz^)) = a — ft and that g(Vi(xaz^)) = a — ft + ei, for any a, ft G N^. The result of the action of the operator in the right-hand side of (3.7) on xaz^ consists of three terms whose grades are a — ft + ui + uj, a — ft + uj and a — ft + ui. Thus the operators Wi, Wj commute if and only if

Qi(D) = (EQi)(D), i,j = 1,...,n, i = j, (3.8)

and

Pi(D)(EUiPj)(D) = Pj(D)(EUjPi)(D), i, j = 1,..., n. (3.9)

It follows from (3.8) that the condition Qi(s + uj) = Qi(s), i, j = 1,..., n, i = j is necessary for the family {Wi}n=1 to be commutative. Under this assumption on the polynomials Qi(s) the compatibility conditions (2.3) can be written in the form

Pi(s + uj)Pj(s) = Pj(s + ui)Pi(s), i, j = 1,... ,n

and they are therefore equivalent to (3.9). The proof is complete.

For x(0) G Cn let Ox(o) be the D-module of formal power series centered at x(0). Let Cx(°)

denote the set of complex numbers C considered as a C[x1,... , xn]-module via the isomorphism

C ~ C[x1,... ,xn]/(x1 — x(0),.. . ,xn — x^). We use the following isomorphism (see Proposition 2.5.26 in [4] or [1], § 4) between the space of formal solutions to M at x(0) and the dual space of Cx(°) ®C[x] M

HomB(M, Ox(°)) ~ Homc(Cx(°) ®C[x] M, C). (3.10)

This isomorphism holds for any finitely generated D-module. Using (3.4) and fixing the point x = x(0) we arrive at the isomorphism

Cx(°) 0C[x] fR[x] / WiR[xA ~ R /± Wi,x(°) R, (3.11)

where Wi)X(o) are obtained from the operators W by setting x = x(0>. Combining (3.10) with (3.11) we see that

n \

J]Wt;x(o) R, C . i=1 )

Thus the following Lemma holds true.

Lemma 3.2 The number of linearly independent formal power series solutions to the system

(1.1) at the point x = x(0> is equal to dim^R/ П=1 Wi x(o) R.

For any differential operator P G D, P = 5^|a|<m c.a(x) its principal symbol a(P)(x, z) G R[x] is defined by a(P)(x,z) = |a|=m ca(x)z“. Let Hj(x,z) = a(Gj)(x, z) be the principal

symbols of the differential operators which define the generalized Horn system (1.1). Let J С D be the left ideal generated by G^...,Gn. By the definition (see [3], Chapter 5, § 2) the characteristic variety char(M) of the generalized Horn system is given by

char(M) = {(x, z) G C2n : a(P)(x, z) = 0, for all P G J}.

Let us define the set UM С Cn by UM = {x G Cn : 3 z = 0 such that (x, z) G Char(M)}. Theorem 7.1 in [3, Chapter 5] yields that for x(0> G UM

HomD(M, Ox(o)) ~ HomD(M, Ox(o)).

It follows from [18] (pages 146,148) that the C-dimension of the factor of the ring R with respect to the ideal generated by the regular sequence of homogeneous polynomials H1(x(0), z),..., Hn(x(0), z) is equal to the product ГШ=1 deg Hj(x(0),z). Since a sequence of n homogeneous polynomials in n variables is regular if and only if their common zero is the origin, it follows that UM = 0 in our setting. Using Lemmas 3.1,3.2, and Lemma 2.7 in [16], we arrive at the following Theorem.

Theorem 3.3 Suppose that the polynomials Pj(s),Qj(s) satisfy the compatibility conditions

(2.3) and that Qj(s + Uj) = Qj(s) for any i,j = 1,...,n, i = j. If the principal symbols H1 (x(0), z),..., Hn(x(0), z) of the differential operators G1,..., Gn form a regular sequence at x(0) then the dimension of the space of holomorphic solutions to (1.1) at the point x(0) is equal to FEU deg H,(x<”> ,z).

Using Lemma 2.7 in [16], we obtain the following result.

Corollary 3.4 Suppose that the principal symbols H1(x(0>, z), ..., Hn(x(0>, z) of the differential operators G1,...,Gn form a regular sequence at x(0>. Then the dimension of the space of holomorphic solutions to (1.1) at the point x(0> is less than or equal to ГЩ=1 degHj(x(0>,z).

In the next section we, using Theorem 3.3, construct an explicit basis in the space of holomorphic solutions to the generalized Horn system under the assumption that Pj,Qj can be represented as products of linear factors and that deg Q > deg Pj, i = 1,..., n.

HomD(M, Ox(o)) ~ Home ( R

4 Explicit basis in the solution space of a sparse hypergeometric system

Throughout this section we assume that the polynomials Pi(s),Qi(s) defining the generalized Horn system (1.1) can be factorized up to polynomials of degree one. Suppose that Pi(s), Qi(s) satisfy the following conditions: Qi(s + uj) = Qi(s) and deg Qi > deg Pi for any i, j = 1,..., n, i = j. In this section we will show how to construct an explicit basis in the solution space of such a system of partial differential equations under some additional assumptions which are always satisfied if the parameters of the system under study are sufficiently general.

Recall that U denotes the matrix whose rows are u1,..., un and let UT denote the transpose of U. Let A = (UT) , let (As)i denote the ith component of the vector As and di = deg Qi.

Under the above conditions the polynomials Qi(s) can be represented in the form

di

Qi(s) = J^[((As)i — aij), i = 1,... , n, aij G C. j=1

By the Ore-Sato theorem [17] (see also § 1.2 of [10]) the general solution to the system of difference equations (2.1) associated with (1.1) can be written in the form

11=111 j=1 r((As)i— aij +1)

where p G N0, ti,ci G C, Ai G Zn and 0(s) is an arbitrary function satisfying the periodicity

conditions 0(s + ui) = 0(s), i = 1,..., n. (Given polynomials Pi, Qi satisfying the compatibility conditions (2.3), the parameters p, ti, ci, Ai of the solution <^(s) can be computed explicitly. For a concrete construction of the function <^(s) see [16]. The following Theorem holds true.

Theorem 4.1 Suppose that the following conditions are fulfilled.

1. For any i, j = 1,..., n, i = j it holds Qi(s + uj) = Qi(s) and deg Qi > deg Pi.

2. The difference aij — aik is never equal to a real integer number, for any i = 1,..., n and

j = k

3. For any multi-index I = (i1,..., in) with ik G {1,..., dk} the product n?=1((Ai, s) — ci) never vanishes on the shifted lattice Zn + 7/, where 7/ = (a1i1,..., anin).

Then the family consisting ofYln=1 di functions

v,(x)=x" y ?+-<■----------------IK..r«A.« + Tt)-ft)------------------------------;r., (42)

. ni n, r((As)t, + ' I. ... - ''; .. + 1)

is a basis in the space of holomorphic solutions to the system (1.1) at any point x G (C*)n =

(C \ {0})n Here Kjj is the cone spanned by the vectors u1,... ,un.

Proof It follows from Theorem 2.1 and the assumptions 2,3 of Theorem 4.1 that the series (4.2) formally satisfies the generalized Horn system (1.1). Let denote the kth row of A. Since deg Qi(s) > deg Pi(s), i = 1,..., n it follows by the construction of the function (4.1) (see [16]) that all the components of the vector A = 5^i=1 Ai — 5^= diXi are negative. Thus for any multi-index I the intersection of the half-space Re(A, s) > 0 with the shifted octant Kj + y/ is a bounded set. Using the Stirling formula we conclude that the series (4.2) converges everywhere in (C*)n for any multi-index I.

The series (4.2) corresponding to different multi-indices I, J are linearly independent since by the second assumption of Theorem 4.1 their initial monomials x71, xYJ are different. Finally, the conditions of Theorem 3.3 are satisfied in our setting since the first assumption of Theorem 4.1 yields that the sequence of principal symbols Hi(x(0), z),... , Hn(x(0), z) G R of hypergeometric differential operators defining the generalized Horn system is regular for x(0) G (C*)n. Hence by Theorem 3.3 the number of linearly independent holomorphic solutions to the system under study at a generic point equals ПГ=1 dj- In this case UM = {x(0) G Cn : x10) ... xi0) = 0}. Thus the series (4.2) span the space of holomorphic solutions to the system (1.1) at any point x(0) G (C*)n. The proof is complete.

In the theory developed by Gelfand, Kapranov and Zelevinsky the conditions 2 and 3 of Theorem 4.1 correspond to the so-called nonresonant case (see [9], § 8.1). Thus the result on the structure of solutions to the generalized Horn system can be formulated as follows.

Corollary 4.2 Let x(0) G (C*)n and suppose that Qj(s + Uj) = Qj(s) and deg Q > deg Pj for any i,j = 1,...,n, i = j. If the parameters of the system (1.1) are nonresonant then there exists a basis in the space of holomorphic solutions to (1.1) near x(0) whose elements are given by series of the form (1.2).

5 Examples

In this section we use the results on the structure of solutions to the generalized Horn system for computing the number of Laurent expansions of some rational functions. This problem is closely related to the notion of the amoeba of a Laurent polynomial, which was introduced by Gelfand et al. in [12] (see Chapter 6, § 1). Given a Laurent polynomial f, its amoeba Af is defined to be the image of the hypersurface f-1(0) under the map (x1,...,xn) ^ (log |x11,..., log |xn|). This name is motivated by the typical shape of Af with tentacle-like asymptotes going off to infinity. The connected components of the complement of the amoeba are convex and each such component corresponds to a specific Laurent series development with the center at the origin of the rational function 1/f (see [12], Chapter 6, Corollary 1.6). The problem of finding all such Laurent series expansions of a given Laurent polynomial was posed in [12] (Chapter 6, Remark 1.10).

Let f(x1,...,xn) = agSaaxa be a Laurent polynomial. Here S is a finite subset of the integer lattice Zn and each coefficient aa is a non-zero complex number. The Newton polytope Nf of the polynomial f is defined to be the convex hull in Rn of the index set S. The following result was obtained in [8].

Theorem 5.1 Let f be a Laurent polynomial. 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 Nf and at most equal to the number of integer points in Nf.

In the view of Corollary 1.6 in Chapter 6 of [12], Theorem 5.1 states that the number of connected components of the complement of the amoeba Af is bounded from below by the number of vertices of Nf and from above by the number of integer points in Nf. The lower bound has already been obtained in [12]. In this section we describe a class of rational functions for which the number of Laurent expansions attains the lower bound given by Theorem 5.1. Our main tool is Theorem 2.1 which allows one to describe supports of the Laurent series expansions of a rational function which can be treated as a solution to a generalized Horn system. In the

following three examples we let u1,... ,un G Zn be linearly independent vectors, p G N and let a1,... , an G C* be nonzero complex numbers. We denote by U the matrix with the rows

of yi(x) with nonempty domain of convergence. These subsets are S0 = {s G Zn : (As)* >

0, i = 1,..., n} and Sj = {s G Zn : v1s1 + ■ ■ ■ + vnsn +1 < 0, (As)* > 0, i = j}, j = 1,..., n. Besides S0,..., Sn there can exist other subsets of Zn satisfying the conditions in Theorem 2.1. (Such subsets “penetrate” some of the hyperplanes (As)* = 0, v1s1 + ■ ■ ■ + vnsn + 1 = 0 without

additional subsets gives rise to a convergent Laurent series and therefore does not define an expansion of y1(x). Indeed, in any series with the support in a “penetrating” subset at least one index of summation necessarily runs from — to to to. Letting all the variables, except for that one which corresponds to this index, be equal to zero, we obtain a hypergeometric series in one variable. The classical result on convergence of one-dimensional hypergeometric series (see [10], § 1) shows that this series is necessarily divergent. Thus the number of Laurent series developments of y1(x) cannot exceed n +1. The Newton polytope of the polynomial 1/y1(x) has n +1 vertices since the vectors u1,... ,un are linearly independent. Using Theorem 5.1 we conclude that the number of Laurent series expansions of y1(x) equals n +1. Thus the lower bound for the number of connected components of the amoeba complement is attained.

(A9)i denote the

u1;..., un and use the notation (A*j) = A = (UT) 1 and v* = A1i + ■ ■ ■ + Ani. The conclusions in all of the following examples can be deduced from Theorem 7 in [14].

Example 5.2 The function y1(x) = (1 — a1xu1 — ■ ■ ■ — araxUn)-1 satisfies the following system of the Horn type

(5.1)

Indeed, after the change of variables xi(^1,..., £n) = ^f14... ^nni (whose inverse is ^ = xui) the system (5.1) takes the form

(% + ■ ■ ■ + + 1) y(C) = %y(C), i = 1,..., n.

(5.2)

The function (1 — a1^1 — ••• — an£n)-1 satisfies (5.2) and therefore the function y1(x) is a solution of (5.1). The hypergeometric system (5.1) is a special instance of systems (5.3) and (5.5). We treat this simple case first in order to make the main idea more transparent.

By Theorem 3.3 the space of holomorphic solutions to (5.1) has dimension one at a generic point and hence y1(x) is the only solution to this system. Thus the supports of the Laurent series expansions of y1(x) can be found by means of Theorem 2.1. There exist n+1 subsets of the lattice Zn which satisfy the conditions in Theorem 2.1 and can give rise to a Laurent expansion

intersecting them; subsets of this type can only appear if | det U| > 1). However, none of these

The function y2(x) = ((1 — a1x“1 — ■ ■ ■ — an-1xUn-1 )p — anxUn) 1 is a solution to the following system of differential equations of hypergeometric type

Indeed, the same monomial change of variables as in Example 5.2 reduces (5.3) to the system

«¿6 Sy(x) = %y(x), i = 1,..., n — 1,

jQ( S + j)j y(x) = (P^in + j^ y(x) (5.4)

where S = 0^ + ■ ■ ■ + 0çn-1 + p0çn + p. The system (5.4) is satisfied by the function ((1 — ai£i — ■ ■ ■ — ara_i£ra_i)p — an^n)-1. This shows that y2(x) is indeed a solution to (5.3). Thus the support of a Laurent expansion of y2(x) must satisfy the conditions in Theorem 2.1. Notice that unlike (5.1), the system (5.3) can have solutions supported in subsets of the shifted lattice Z”+y for some y G (0,1)”. Yet, such subsets are not of interest for us since we are looking for Laurent series developments of y2(x). The subsets S0 = {s GZ” : (As)* > 0, i = 1,..., n} and Sj = {s G Z” : (As)i + ■ ■ ■ + (As)„_i + p(As)„ + p < 0, (As)* > 0, i = j}, j = 1,..., n satisfy the conditions in Theorem 2.1. The same arguments as in Example 5.2 show that no other subsets of Z” satisfying the conditions in Theorem 2.1 can give rise to a convergent Laurent series which represents y2(x). This yields that the number of expansions of y2(x) is at most equal to n +1. The Newton polytope of the polynomial 1/y2(x) has n +1 vertices since the vectors ui,... ,U” are assumed to be linearly independent. Using Theorem 5.1 we conclude that the number of Laurent series developments of y2(x) equals n +1.

Example 5.4 Let H be the differential operator defined by H = p(A0)2 + ■ ■ ■ + p(A0)n + p. Using the same change of variables as in Example 5.2, one checks that y3(x) = ((1 — aixui)p — a2x“2 — ■ ■ ■ — anx“n)_i solves the system

aixui ((A6l)i + H) y(x) = (A0)i y(x),

OiXUi±K ( h ((A0)i + ’K + j) j y{x) =

\j=0 J (5.5)

W* ^I1(H — p + j)j y(x) i = 2,...,n.

Analogously to Example 5.2, we apply Theorem 2.1 to the system (5.5) and conclude that the number of Laurent expansions of y3(x) at most equals n + 1. Thus it follows from Theorem 5.1 that the number of such expansions equals n + 1.

Example 5.5 The Szego kernel of the domain {z G C2 : |zi| + |z2| < 1} is given by the hypergeometric series

F(2si + 2s2 + 2) S2 _

>0 r(2si + 1)r(2s2 + 1)

Sl,S2>0

(1 - Xi - x2)(1 + 2xix2 - x? - + 8xix2

-----------------------------;-----------2------------• l^.o)

((1 - X? - X2) - 4X1X2)

(See [2], Chapter 3, § 14.) This series satisfies the system of equations

Xi (20i + 202 + 3) (20i + 202 + 2) y(x) = 20t(20t - 1)y(x), i = 1, 2.

There exist three subsets of the lattice Zn which satisfy the conditions in Theorem 2.1, namely {s G Z2 : si > 0, s2 > 0}, {s G Z2 : si > 0, si + s2 + 1 < 0}, {s G Z2 : s2 > 0, si + s2 + 1 < 0}. Using Theorem 2.1 we conclude that the number of Laurent expansions centered at the origin of the Szego kernel (5.6) at most equals 3. The Newton polytope of the denominator of the rational function (5.6) is the simplex with the vertices (0, 0), (4, 0), (0, 4). By Theorem 5.1 the number of Laurent series developments of the Szego kernel at least equals 3. Thus the number of Laurent expansions of (5.6) (or, equivalently, the number of connected components in the complement of the amoeba of its denominator) attains its lower bound.

Example 5.6 Let u? = (1, 0),u2 = (1,1) and consider the system of equations

) m-r).

xU2y(x) = )y(x).

The principal symbols H?(x, z), H2(x, z) G R[x] of the differential operators defining the system (5.7) are given by H1(x,z) = -x?z? + x2z2, H2(x,z) = -x2z2. By Theorem 3.3 the dimension of the solution space of (5.7) at a generic point is equal to 1 since dime R/(H?(x, z), H2(x, z)) = 1 for x?x2 = 0. For computing the solution to (5.7) explicitly we choose 7 = 0 and consider the corresponding system of difference equations

^(s + ui)(si - s2 + 1) = ^(s), (58)

<^(S + U2)(S2 + 1) = ^(s). ( . )

The general solution to (5.8) is given by <^(s) = (r(s? - s2 + 1)r(s2 + 1))-10(s), where 0(s) is

an arbitrary function which is periodic with respect to the vectors u?,u2.

There exists only one subset of Z2 satisfying the conditions of Theorem 2.1, namely S = {(s?, s2) G Z2 : s? - s2 > 0, s2 > 0}. Choosing 0(s) = 1 and using (4.2), we obtain the solution to (5.7):

Si S2

/v* 1 /y> 2

y(*)= E r(8,-82 + l)rfe+l)=eXP(j№+^ (5-9)

Si — S2 > 0,

S2 > 0

It is straightforward to check that the solution space of (5.7) is indeed spanned by (5.9).

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РАЗРЯЖЕННЫЕ ГИПЕРГЕОМЕТРИЧЕСКИЕ СИСТЕМЫ

Тимур Садыков Сибирский федеральный университет,

пр. Свободный, 79, Красноярск, 660041, Россия, e-mail: [email protected]

Аннотация. Описывается подход к изучению теории гипергеометрических функций от нескольких переменных с помощью обобщенной системы дифференциальных уравнений типа Горна. Получена формула для вычисления размерности пространства решений этой системы, основываясь на которой строится в явном виде базис ее пространства голоморфных решений при некоторых ограничениях на параметры системы.

Ключевые слова: гипергеометрические функции, системы дифференциальных уравнений типа Горна, система Меллина.