Domains of convergence for A-hypergeometric series and integrals Текст научной статьи по специальности «Математика»

YflK 517.55+512.761

Domains of Convergence for A-hypergeometric Series and Integrals

Lisa Nilsson* Mikael Passare^

Stockholm University Stockholm Sweden

*

August K. Tsikh*

Institute of Mathematics and Computer Science Siberian Federal University Svobodny, 79, Krasnoyarsk, 660041

Russia

Received 17.05.2019, received in revised form 10.07.2019, accepted 15.08.2019

We prove two theorems on the domains of convergence for A-hypergeometric series and for associated Mellin-Barnes type integrals. The exact convergence domains are described in terms of amoebas and coamoebas of the corresponding principal A-determinants.

Keywords: A-hypergeometric series, Mellin-Barnes integral, r-integral, Principal A-determinant. DOI: 10.17516/1997-1397-2019-12-4-509-529.

1. Introduction and preliminaries

The hypergeometric functions constitute a substantial section within the realm of special functions of Mathematics and Physics. It may therefore appear quite surprising that this wide class of hypergeometric functions in essence is obtained from a single "exponential" series by restriction of the summation to a suitable sublattice.

We rewrite the exponential Taylor series as a Laurent series

using the fact that the Euler gamma-function r has poles in negative numbers. Now consider the shifted exponential series

with y = (ji,..., yn) G CN, and introduce the notion of T-series.

* lisa@math.su.se

1 passare@math.su.se ^ atsikh@sfu-kras.ru © Siberian Federal University. All rights reserved

Definition 1. Let L be an integer sublattice of ZN. The diagonal subseries

№ = (a) = £ r(7l + + 1)... p(7n + In + 1) (1)

is called a T-series.

This definition was introduced by Gelfand, Kapranov and Zelevinsky [1]. They observed that choosing bases of L, one can rewrite any T-series as a product of a monomial a? with a formal Laurent series which is a hypergeometric series as defined by Horn ([2]) in m variables, where m = rank(L). Recall that a Laurent series is hypergeometric in the sense of Horn if the quotients of neighbor coefficients are rational functions in the variables of summation. Usually such coefficients are represented by ratio of T-factors, i.e. of factors of gamma-function r composed with affine function in the variables of summation.

In the mentioned paper [1] a new fruitful approach to the general theory of hypergeometric functions was developed. It has connections to toric geometry, combinatorics of polytopes and a number of other fields. The basic idea of the GKZ-approach is to cleverly introduce extra variables aj, one for each T-factor in the hypergeometric series in the sence of Horn. Their main observation was that the new function (of many variables) thus obtained will satisfy a very simple (binomial) system of differential equations with constant coefficients.

The corresponding hypergeometric system of differential equations, see Definition 2 in the Section 2, may be coded by an integer matrix A of size n x N, where n = N — m, whose kernel sublattice is L (recall that m = rankL).

Note that a (N x m)-matrix B as an annihilator of A is called the Gale transform of the configuration A, see Section 5.4 of [3]. We are interesting in the Gale transforms B of two types. The first type consist of integer matrices whose columns generate the lattice L. The second one contains rational matrices B which have a unit matrix Em on some m rows of B.

We define the hypergeometric series associated to rational Gale transform B with a unit matrix on the last m rows as follows:

al+(B,k)

j>(a) = ^b (a)=Y -, (2)

¿NmjT r(Yj + (bj ,k) + 1) k!'

where the bj denote the first n rows in B, k! = k 1!... km!, and

a?+(B,k) = ay (abl )k> ... (abm )k™

with b1,... ,bm being the columns of B. In fact series (2) is a sum of r-series (1) with suitable shifts

It is well known that the hypergeometric series in one variable can also be represented as a so called Mellin-Barnes integral. Given the hypergeometric series (2), its formal integral representation is the following Mellin-Barnes integral (about the multiple Mellin-Barnes integrals see [4])

4>(a) = T^- ( N r(Sl) ... r(Sm)-TT (—abj )-'j ds,

^ ) (2ni)T Js+iKmUN=-T r(jj — (bj,s) + 1) ii( ) ,

IN-m -

j=1 r(j — (bj ,s) + 1) j=1

where ds = ds\ A ■■■ A dsm and 6 € R!p is chosen in appropriate way. This means that granted that the integral < converges, it will coincide with series (2), where the convergence domains of the integral and the series overlap.

In this paper we also introduce another type of integrals called r-integrals, as the continuous analogue of the T-series (1):

<(a) = = i - 7)aY-dM (3)

JhcL c

and call it a r-integral. Here h is a cycle of dimension m = N — n with a closed support, d^ is an m-dimensional measure on LC, and

N

T(£ — 7)aY-i = IIr& — Yj)a]j. j=i

Notice that we have put all T-factors in the numerator. This will turn out to be very beneficial since it will make the domain of convergence larger.

The aims of this paper is to specify the convergence domains for the series (8) which is in fact the dehomogenization of the series (2). We will use the theory of amoebas (see Definition 5), and the so called principal A-determinant, see Definition 3, which defines the singularities of the hypergeometric functions. Parallely, we will be studying the representation of the hypergeometric function in terms of Mellin-Barnes type integrals. We will describe the convergence domains of these integrals and their connection with the coamoeba of the principal A-determinant.

These results were largely obtained already in 2009 and included in the thesis by L. Nilsson [5], advised by M. Passare and A. Tsikh, but have not been published until now. The years since then have shown an increasing interest to them from specialists in hypergeometric and algebraic functions. Recall that already H. Mellin noticed [6] a general algebraic function y(x) = y(xi,... ,xn-l) is a hypergeometric function. The convergence domains for the series and integrals representing this function were described in the papers [7] and [8], correspondingly. Using this results, in [9] the monodromy of y(x) was described. We hope that Theorems 2,3,4 of this paper can be applied to give a similar description of monodromy for an arbitrary hyper-geometric function.

2. Basic definitions and algebraic preparations

2.1. A-hypergeometric systems

Gelfand, Kapranov and Zelevinsky constructed a holonomic system of linear differential equations satisfied by the Horn series as well as their integral representations of Euler type generalizing the integral representation for 2F1 in [1, 10]. This is done through associating a system of hypergeometric functions to a finite subset A of the integer lattice zn. These functions are called A-hypergeometric functions.

The set A has to satisfy the following conditions: the z-span of A is zn and there exists a linear form h with the property h(a) = 1 for all a G A. As mentioned in Section 1 we choose h as a coordinate function and therefore we are able to represent A as a matrix

A = (a11) a12) ! ! ! a1 )) ' (4)

and sometimes we will identify A with a set of vectors

A = {a(1),...,a(N>}C Zn-1.

Denote by L = L(A) c za the lattice of affine relations among elements of A, that is, the set of integer vectors l = (la)aeA, such that

y^ laa = 0.

aeA

Let ca be the space of vectors a = (aa)aeA. Corresponding to the order of the set A we also denote the elements a G ca by a = (a1,..., aN). For any l = (l1,... ,lN) G L we define the differential operator □l on ca by

□i = n (d/daj)lj — n (d/daj —. j:lj >0 j:lj <0

We also define the differential operators

N

Ei = ^^ a(j)aj (d/daj), i = 1, ... ,n, j=1

on ca, where a(j) is the i:th coordinate of the j:th column of A.

Definition 2 ([1]). Let ft = (ft1,..., ¡3n) be a complex vector. The A-hypergeometric system with parameters ft is the following system of linear differential equations on a function $(a), a £ CA:

□l$(a) = 0, l G L

Ei$ — fti<b = 0, i = 1,... ,n. (5)

Hence the A-hypergeometric system consists of firstly a binomial equation in partial differentials given by □l and secondly of a number of a equations given by Ei that should really be considered as a number of homogeneity conditions, originating from the relations among the elements of A.

Remark 1. Formally □l$ = 0 with l G L in (5) represents an infinite number of equations, but in fact it is enough to consider a finite number of equations with l corresponding to a basis of the lattice L.

Remark 2. One can check immediately (see [1]) that any r-series (1) with 7 £ A-1(ft) formally satisfies the A-hypergeometric system (5).

Definition 3 ([1]). Holomorphic solutions of (5) will be called A-hypergeometric functions.

It turns out that the solution space to the system in Definition 2 is finite dimensional, more precisely, we have the following theorem.

Theorem 1 ([1, 10]). The number of linearly independent holomorphic solutions of the A-hypergeometric system (5) at a generic point of ca is equal to the normalized volume of the convex hull Q(A) of the points in A.

The volume is normalized so that the minimal (n — 1)-simplex in rn-1 with vertices on zn-1 has volume 1.

In [1] it is also proved that choosing corresponding vectors y one can construct a basis of solutions to (5) by means of nonformal (convergent) Г- series (in more details see subsection 3.1).

2.2. The principal A-determinant, and the notions of amoeba and coamoeba

The solutions to the A-hypergeometric system define regular functions everywhere in ca except on the algebraic hypersurface, {EA(aa) = 0}. The description of the defining polynomial is the following.

Definition 4 ([11]). Let A С zn-1 be a finite subset which affinely generates zn-1. For any f = f(x1 ,...xn-i) £ ca, where ca is interpreted as the space of Laurent polynomials with monomials from A, the principal A-determinant is defined as a resultant

EaU ) = Ra(X1 , ■ ■ -,xn~1 X

Note that Ea is clearly a polynomial function in coefficients of f. One major aim of this paper is to describe the exact convergence domains of the solutions to the A-hypergeometric system. However the solutions can be represented in various different ways, such as power series and different types of integrals. The different representations will be solutions to the same hypergeometric system, but have different convergence domains. They are in fact analytic extensions of each other. Naturally the convergence domains of the hypergeometric power series only depend on the absolute value of the variables, which leads us to the key tool for describing the situation further, that is, amoebas: For a Laurent polynomial P in m variables, we denote by ZP the hypersurface determined by the equation P = 0, and introduce the logarithmic mapping (c*)m ^ rm given by

Log : (C1,...,Cm) ^ (log|C1|, ■■■, log I Cm I) ■

Definition 5. The amoeba of the polynomial P is denoted by Ap and

Ap := Log(Zp).

The amoebas properties are discribed in the papers [12, 13] and [14]. We also consider solutions represented as so called Mellin-Barnes integrals or Г-integrals. The convergence of the integrals depend only on the argument of the variables, and hence it is natural to use coamoebas in the study of the convergence domains of these integrals.

Similarity to the definition of an a amoeba above, we use the argument map for defining coamoebas. Arg : (c*)m ^ rm/[0,2n]n given by

Arg : (C1, ■■■,Zm) ^ (arg(Z1), ■ ■ ■,arg(Zm)).

Definition 6. The coamoeba is denoted by A'P, where

A'p := Arg(Zp).

It is convenient to incorporate the maps Log and Arg into the following diagram where p is a projection on the m-dimension real torus:

Log

(c*)m

Exp '

Arg

rm ^-cm-r"

Re Im

2.3. Algebraic preparations

Proposition 1. Let A be a matrix of size n x N, n ^ N with integer entries. Then the following claims are equivalent:

(i) the column span of A is the entire lattice zn;

(ii) the maximal minors of A are relatively prime;

(iii) there is a unimodular matrix M of size N x N such that AM = (En\0) (the unity matrix of size n x n enlarged by adding zeros to a n x N-matrix);

(iv) there is a completion A which is unimodular, i.e. we can enlarge A to a N x N integer square matrix A with det A = 1.

Proof. (i) ^ (ii) Given the condition (i), there exists an integer Nxn-matrix C with the property that AC is equal to the unit matrix En of size n x n. From this it follows that all the maximal minors of the matrix A are relatively prime, since by the well known Binet-Cauchy formula ([15]) the determinant of AC equals the sum of the maximal minors of A multiplied with the corresponding minors of C.

(ii) ^ (iii) By the invariant factor theorem ([16]) it follows that there exist unimodular integer matrices D and F of size n and N respectively, such that DAF = (S\0) where S is a diagonal n x n-matrix with integers e1,... ,en on the diagonal, and 0 is the zero-matrix of size n x (N — n). From the representation A = D-1(S\0)F-1 it is now easy to see that in fact S = En, because some ej being different from 1 would contradict the fact that the maximal minors of A are relatively prime.

(iii) ^ (iv) By (iii) we have

A = D-1(En\0)F-1 = (D-1\0)F-1, and the desired enlargement of A may be taken to be

A:

(D-'\ 0 )

V 0 En-n )

F -

(iv) ^ (i) This is obvious. □

We now introduce some notation. Given an integer (n x N)-matrix A, denote by B the integer dual matrix of A, i.e. a (Nx m)-matrix such that the columns form a z-basis for the kernel of A. For any increasingly ordered subset I = (i17... ,in) of {1, 2,..., N), we let J = (j1,... ,jm) be its complement, also increasingly ordered. We let Aj denote the (n x n)-minor of A with columns indexed by I, and similarly, we write Bj for the (m x m)-minor of B with rows having indices from the complementary subset J.

nz)"

p

1

Proposition 2. Assume that columns of A generate the entire lattice zn. Then | det Aj| = = | det BjI.

Proof. We can assume that I = {1,... ,n} and J = n + 1,..., N. So we can write A in the block form (Aj | AJ). By analogy with respect to rows we write the matrix B in block form

B

J

Since the columns of A generate zn it follows from Proposition 1 that there is a unimodular completion A of A, which we write in the block form

i Aj A^ I * * I

A =

Let us consider the corresponding block composition of the inverse matrix :

B'

A-1 =

H B'j\

According to Jacobi's formula ([15], formula (11)) one has det BJ ■ detA = det Al and hence | det BJ | = | det Al |. It is clear that the block column

J

in A 1 constitute a basis of L, so BJ differs from BJ only by an unimodular factor. Finally we get | det BJ | = | det BJ | = | det Al ^ □

3. Domains of convergence for A-hypergeometric series 3.1. A-hypergeometric series

Let us firstly explain the GKZ-approach of the constructing basis A-hypergeometric series for the A-hypergeometric system (5). Given the data for the system(5), that is, an integer (n x N)-matrix A of type (4), and a generic complex column n-vector ¡3, we fix a choice of a basis of the sublattice L = KerA. This means that we choose an integer (N x m)-matrix B, where m = N — n, such that AB = 0 and the columns of B form a z-basis for L.

Using the row vectors bj of the matrix B and a complex column vector y € A-1(3), we can rewrite the T-series (1) on the form

N aYj + (bj ,k)

(a) = E II r(1+j + b u\) • (6)

ktZm jil r(l+ Y + b ,k))

Due to Remark 2 the series (6) gives a formal solution to the A-hypergeometric system provided that the vector y is chosen so that Ay =

In order to obtain convergent T-series, in [1] was suggested to choose the vector y so that m of its entries are integers. The point is that when Yj € z the factor r(1 + Yj + (bj, k)) will be infinite for all integer k in the halfplane 1 + Yj + (bj,k) ^ 0, so the coefficients in (6) are

zero for such k. With m different entries (Yj1,..., Yjm) = ■ YJ of Y being integers, the support of summation in the series (6) will be contained in a simplicial cone, if the matrix Bj with rows (bj1,..., bjm) is nondegenerate, and one gets a series with a non-empty domain of convergence.

According to Proposition 2 we have \ det Aj \ = \ det BJ \, and we denote this number by Sj. Now, the choice of I, that is, the choice of n columns of the matrix A, is of course equivalent to the choice of a subsimplex a = aj of the point configuration A C zn-1 c rn-1. One clearly has Sj = (n — 1)! Vol (aj).

The objective is now, for each subset I, to first construct Sj linearly independent convergent r-series, and then to determine their common domain of convergence. Of course we only have to consider those I for which Sj = 0, so that the minors Aj and BJ are invertible. In the linear compatibility equation Ay = ¡3 for y = (yi ,YJ), we can then solve yj in terms of ¡3 and yj . This leads to the identity

Y = Cfi + BB-1Yj (7)

where C is the (N x m)-matrix obtained from A-1 by inserting zero rows corresponding to the indices from the complement J. We have here made use of the relation A-1AJ = —BjB-1, which follows from the equation 0 = AB = AjBj + AJBJ.

From equation (7) we see how the vector y that defines the series (6) is computed from the shorter vector yj . The algorithm used by GKZ in [1] is to take all choices of integer vectors YJ such that each entry of the vector B-1yj belongs to the half-open interval [0,1). There are precisely Sj distinct such choices of YJ, each giving a different vector Y by (7), and hence producing a different series (6).

Recall that due the Definition 7 a general hypergeometric series we defined for every rational Gale transformation R of the form (R', Em)tr and for y' & cn, to be the following power series

al' + (R,k)

6(a) = 6ry' (a) = > —¡r=-, (8)

' ' ¿kmUN-r r(Yj + (j k + 1) k! y>

where the rj denote the rows in the matrix R', and k! = k1! • • • km!.

Remark that up to the factor aY one can consider (8) as a power series with the exponents from the lattice Rzm. Clearly, R = BB-1 for each basis B of L, so L is a sublattice of Rzm, and hence (8) is a finite sum of r-series where Y(t) runs over the Rzn/L. In example

above one has RX = (1 b1,b2) with unimodular matrix X = ^ ^, and it means that

Rz2 = (RX)z2 = LU (1 b1, b2). So we get y(1) = 0 and y(2) = 1 b1 like above in the GKZ-approach. In general we have the following statement.

Proposition 3. For a given integer n x N-matrix A of the type (4), whose columns generate zn, and a chosen simplex a C Q of normalized volume A, there will be precisely A distinct A-hypergeometric series of type (8) which are linearly independent.

Proof. Let B be a basis of L = A-1(0). Consider the lattice M ■= Rzm = BB-1zm. By the invariant factor theorem there are unimodular m x m-matrices X and Y, such that the new bases for the lattices L and M given by

R = RX, B = BY

have the property that the basis B is expressed in the basis R by means of a diagonal integer matrix:

/ ¿1

B = R-

Sj e Z.

\ Sm

In other words, if b1,... bm denote the column vectors of B, then the lattice M is Q-generated by the basis B of L as

,. [b)1 bm 1 M =1 —si + ... + — sm\ .

\ ) sezm

From this it is easily seen that the series (8) can be re-written in the powers

(a^)S1 ... (ahm/g-)Sm

Clearly, the index M : L is equal to A = | det R''\ = ... 5m\ and by choosing various radicals (abb )1/Sj we obtain A linearly independent series. □

We now introduce a dehomogenization of the series (8), where all other variables than the ones chosen by the position of the unit matrix equals 1, that is a1 = ... = an = 1 and an+k = zk, k =1.. .m.

Definition 7. For every Gale transformation R = (R',Em)tr, we define the dehomogenized hypergeometric series in m variables

j>(z) = b (z)= V ---, (9)

¿mjT r(Yj + (r3k + 1) k! ' K>

where the rj denote the rows in the matrix R', zk = z11 ... zm and k! = k1!... km!.

Notice that the series in the Definition 7 are represented as Taylor series, due to the fact that we have chosen the matrix R to be on the form (B ', Em)tr, with the unit matrix positioned in the last m rows. However by performing monomial changes of variables (9) represents more general series, so called Laurent-Puisieux series.

In fact, there is a natural correspondence between the following actions:

• Choosing a (n — 1)-simplex a C conv(A), i.e. in the Newton polytope of A, which we denote Q = Q(A).

• Choosing a set of n linearly independent column vectors in the matrix A.

• Choosing a set of n rows rj in the dual matrix R, such that the remaining m rows in R give the unit matrix.

The fact that the chosen position of the unit matrix in R also corresponds to a certain choice of simplex in Q will play an important role when considering the convergence of the above series further into this paper.

3.2. Domains of convergence for the hypergeometric series

We want to find the convergence domain of the series defined in Definition 7. We prove the following result, where Aa is the amoeba of the reduced principal determinant EA(1,a'').

Theorem 2. The domain of convergence Da of the series 0(1, a'') in (8) is a complete Reinhardt domain with the property that the corresponding convex domain Log(Da) contains all the connected components of the amoeba complement rm \ Aa, that are associated with the triangulations of (Q, A) containing the simplex a (i.e. are associated with a certain vertex in the secondary polytope £(A)), while it is disjoint from all the other components.

In order to prove this result we want to construct a triangulation of (Q, A), i.e. a triangulation on Q with the set of vertices on A. We do this in the following way. Take any function ^ : A ^ r and consider in the space r"+1 = r" x r the union of the vertical half-lines

{(u,y) G A x r : y < ^(u)}.

Let be the convex hull of all these half-lines. This is an unbounded polyhedron projecting onto Q. The faces of G^ which do not contain vertical half-lines (i.e. are bounded) form the bounded part of the boundary of G^, which we call the upper boundary of G^. Clearly the upper boundary projects bijectively onto Q. If the function ^ is chosen to be generic enough, then all the bounded faces of G^ are simplices and therefore their projections to Q form a triangulation of (Q, A).

Let T be an arbitrary triangulation of (Q, A), and let ^ : A ^ r be any function. Then there is a unique T-piecewise-linear function g^,T : Q ^ r such that g^,T(u) = ^(u) when u is a vertex of the triangulation T. The function g^,T is obtained by affine interpolation of ^ inside each simplex. Note that the values of ^ at points that are not vertices of any simplex of T does not affect the function g^,T.

Definition 8. Let T be a triangulation of (Q, A). For each simplex a of this triangulation we shall denote by C(a) the cone in ra consisting of functions ^ : A ^ r with the following properties:

(a) the function g^,T : Q ^ r is concave.

(b) for any u G A which is not a vertex of the simplex a in the triangulation T, we have g^,T(u) > ^(u).

Now, let A C z"-1 be a finite subset, and Q the convex hull of A as before. We assume that dim(Q) = n — 1. Fix a translation invariant volume form Vol on r". Let T be a triangulation of (Q, A). By the characteristic function of T we shall mean the function T(u) : A ^ r defined as follows:

Vt (u) = ]T Vol(S),

S:ueVert(S)

where the summation is over all (maximal) simplices 6 of T for which u is a vertex. In particular, VT(u) =0 if u is not a vertex of any simplex of T. Let ra denote the space of all functions A ^ r.

Definition 9. The secondary polytope £(A) is the convex hull in the space of the vectors vt for all the triangulations T of (Q, A).

The normal cone to £(A) at every yT will be called NVT £(A) and consists of all linear forms 0 on ra such that

0(yT)= max 0(if). ves(a)

The point yt is a vertex of £(A) if and only if the interior of this cone is non-empty. The union of the normal cones NVTi £(A),..., NVTk £(A) where T1,... ,Tk are all the triangulations of (Q, A) that contains the simplex a will be called the normal cone X(A).

Proof of the theorem 2. We know from the implicit function theorem that Da is not empty, and Abel's lemma [17] tells us that whenever a point a belongs to Da, then so does the full polydisc centered at a. Therefore Da is indeed a complete Reinhardt domain , and the corresponding domain Log(DCT) will contain the negative orthant — r+-1 in its recession cone C(a).

In fact, we will show that C(a) is the negative orthant. This we can see by letting the function 0 = 0 on all the points a(j) in the simplex a. This corresponds to chosing exactly this simplex a. (We could choose 0 equal to anything at the points in a.) Now C(a) consists of functions 0 : A ^ r such that is concave and (w) < 0(w) for all w which are not vertices in a. Hence 0(w) < 0 for all w and all functons 0, and thus C(a) is equal to the negative orthant r+-1.

Let E be a connected component of rn-1 \ Aa that intersects the domain Log(DCT). Then we claim that we actually have an inclusion E C Log(DCT).

It follows, from what we have prooved so far, that the domain Log(DCT) cannot intersect any component of the amoeba complement rn-1 \ Aa whose cone C(a) is not in the negative orthant. On the other hand every connected component of rn-1 \ Aa with the corresponding cone C(a) contained in r+—1 will necessarily intersect, and hence be contained in the domain Log(DCT). The following proposition therefore suffices to make the proof of Theorem 2 complete. □

Proposition 4. The normal cone NVT £(A) at a vertex of the secondary polytope is con-

1

H

tained in the negative orthant — r+ 1 if and only if the corresponding triangulation of Q contains

the simplex a. In fact the union of such normal cones NVa£(A) is equal to — r'

+—1

Proof. We will prove this proposition by proving that the normal cone £(A) coincides with the cone C(a). See ([11]). We get at once from the definitions of yT and g^,T, and the fact that the integral of an affine-linear function g over a simplex a is equal to the arithmetic mean of values of g at the vertices of a times the volume of a, the following ([11], Ch. 7):

(0,^t) = n/ g^,T(x)dx. (10)

JQ

We now fix 0 e Ra. The upper boundary of G^ can be regarded as the graph of a piecewise-linear function g^ : Q ^ r.

g^(x) = max{y : (x,y) e G^}.

We can furthermore state about the function g^ the following:

(a) g^ is concave.

(b) For any triangulation T of (Q, A) we have g^(x) > g^,T(x), Vx e Q.

(c) We have

max (0, y) = n / g^ (x)dx. (11)

vez(A) Jq

(a) follows by construction. To varify (b), it suffices to consider x varying in some fixed simplex a of T. By definition, is affine-linear over a and g^(w) > 0(w) = g^,T for any vertex w G a. Hence the inequality is valid over a. The maximum in (11) can be taken over the set of the yT for all triangulations T of (Q,A), since £(A) is defined as the convex hull of these yT. Hence part (b) together with (10) imply that the left hand side of (11) is greater than or equal to the right hand side. To show the equality, it suffices to exhibit a triangulation T for which g^ = g^,T. To do this, we consider the projections of the bounded faces of the polyhedron G^ into Q. These are polytopes with vertices in A. Take a generic 0' close to 0. Then the bounded faces of the polyhedron G^ give a triangulation T of (Q, A) which induces a triangulation of each of the above polytopes. Hence g^ is T-piecewise-linear and coincides with g^,T. This proves (11). Hence the cones coincide. □

Remark 3. Theorem 2 was proven for the special case n = 2 in [7].

4. Mellin-Barnes integrals

4.1. General Mellin-Barnes integrals and their domains of convergence

By the multiple Mellin-Barnes integral we mean the integral

p

n r((aj,z) + cj)

^ = (2^ / -1 ••• mm ^ (12)

5 + iRm fi r((bk ,Z) + dk )

k = l

where all vector parameters aj ,bk £ rm are real, the scalar parameters cj ,dk £ c, and ds = = dsi... dsm. The vector S £ rm is chosen so that the integration subspace S + irm is disjoint from the poles of the gamma-functions in the numerator. For brevity we rewrite (12) as

^=(2^ / F (s)z-SdS' (13)

denoting by F(s) the ratio of the products of gamma-functions, and z-s denotes the product z-Sl ■ ■ ■ zm-Sm. We suppose that the variable z varies on the Riemann domain over the complex torus tm = (c \ 0)m, so we assume that

z-Sj = e-sjlog , arg zj £ r.

We introduce the following notations:

Xj = Resj, yj = Imsj, j = 1,... ,m.

Let x and y be the vectors in rm with coordinates Xj and yj, correspondingly. Denote by d = Argz = (arg z1,..., arg zm), and

g(y) = E,y)\-E\(bk ,y)\. j=1 k=1

Theorem 3. For any S + irm outside polar sets, the convergence domain of the Mellin-Barnes integral (12) is equal to Arg-1(U°) where U° is the interior of the set

7T

■ I (y^W <

2'

U = p| {e & rm ■ \(y,e)\ < 2g(y)}.

\\y\\=1

In the case when U° = 0 it coincides with the interior P° of the polytope

7T

p = {e & rm ■ \(vv,e)\< ^g(vv), v = i,...,d}

where ±v1,..., ±vd is the set of all unit vectors which generate the fan K corresponding to the decomposition of rm by hyperplanes

(aj,y) = 0, j = 1,...,p and (b2,y) = 0, k =1,...,q.

Proof. Let u,v & r. Since the asymptotic equality

\v\u-1/2 - (\v\ + 1)u-1/2a,s\v\^TO

is valid for every fixed u & r, Stirling's formula implies that there are constants C1 and C2 such that

C1(\v\ + 1)u-1/2e-nH < \r(u + iv)\ < C2(\v\ + 1)u-1/2e-n\"\, (14)

where u & K C r \ {0, —1, —2,...} (K is a compact set), v & r, and the constants C1 and C2 depend only on the choice of K. Using (14), and our notation vj = Imsj, we can make the following estimate for the integrand in (13):

FTP

\F(s)z-s\< const^^ exH(y,e) — 2g(y)} (15)

where

j = (\a,y)\ + 1)a'x)+°j-1/2, & = (\(bk,y)\ + 1)(bk,x>+dk-1/2, and g(y) was defined above. Moreover, (15) holds for all y & rm and all x in compact subsets of r" disjoint from the polar hyperplanes

{(aj, x) + Cj = —v}, {(bk, x) + dk = —v}, v = 0,1, 2,...;

in particular, (15) is valid for x = S. It follows from (15) that, for each e = Argz, provided the inequality

/ m p \ (y,e) < 2 £ \(a3 ,y)\ —Y,\(bk ,y)M for any y & rm \ {0} (16)

2 j=1 k=1

the integrand in (13) decreases exponentially as ||y|| ^ to. Therefore for such e the integral (13) converges absolutely. By homogeneity, (16) is valid for all y & rm \ {0} whenever it holds for y on the sphere { y ■ | y| = 1 } . It means that this integral converges for all e = Argz from the intersection of halfspaces:

p {e ■ (y, e) <2 g(y)}.

\\y\\ = 1

The unit sphere \\y\\ = 1 is symmetric relative to the origin. Since g(—y) = g(y) this implies that the mentioned intersection of halfspaces coincides with the intersection

u ■= p {e ■ \(y,e) \ <2 g(y)}

\\g\\=1

of strips.

It is clear that the integral (13) does not converge for 0 = arg z outside of the closure U, since

the estimate in (14) implies not only (15), but for some other constant the reversed inequality:

p

= -

\F(z)z-s\ > const exp {(y,0) - 2g(y)}- (17)

n 6 2

k=1

- n

Thus if 0 € R— \ U we have an inequality (0,y) > 2g(y) for some y on the sphere ||y|| = 1. By

(17) it means that the integrand in (13) and (12) increases exponentially in some open cone of R—, and therefore is not integrable. Hence Arg-1(U) coincides with the doamin of convergence of the integral (12).

Of course, U c P°. Let us explain why any point 0 € P° belongs to U. Indeed g(y) is a piece-wise linear function whose graph has corners only over the hyperplanes (a-, y) =0 and (bk,y) = 0.

Correspondingly, the function (y) := (0,y} is a piece-wise fractional linear function with

g(y)

respect to the variable y. Consequently all extremal points of the function ' g (y) = (y)

\\y\\ = i

lie on the vertices set of the spheric polyhedron K n {||y|| = 1}, i.e. on the set {±w1,..., ±vd}. Therefore using again the property g(-y) = g(y) we get that 0 € P° implies 0 € U. □

Remark 4. Some partial results on the domains of convergence for integral (12) were obtained in [8, 18, 19], and [4].

4.2. Reduction of hypergeometric series to Mellin-Barnes integrals

Given the hypergeometric series (2) with = aY(ab )kl • • • (ab )km, its formal integral

representation is the following Mellin-Barnes integral

<(a) = -¡—Y>— i N r(Sl) ^ r(Sm) (-ab1 )si • • (-abm ) —ds,

(2nl)— J N-— r( b \ + U ) )

S+iRm H i(Yj - (bi,s) + ^ j=l

for some appropriately chosen S € R—. This means that granted that the integral < converges, it will coincide with < in (8), where the convergence domains of the integral and the series overlap. Let ReY- = c- and choose c- such that the polyhedron

n := {xl > 0,l = 1,..., m; cj - (bj, x) < 0,j = 1,..., N - m}

becomes a simplicial m-dimensional polytope. Choose S in the interior of n. Using the equation

2ni

r(Z )r(1 - Z)

einZ — e-inZ

we get

where

<>(a) = l2r— IF(si) II - (s)(-abl)-S1 ••• (-abm)-Sm ds,

(2ni) JS+iRm l=1 -=1

rj(s) = (2ni)—-NT(-Yj + (bj, z))(ein('b> - e-in('b)).

N-m

Since bj = ( — 1,..., —1) we have that for any division of { 1, ... , N — m } into two

j=i

groups I and J, we get

Y^bi — E bj = ( —1,..., —1) — bj.

iei jeJ jeJ

This implies that

N-m

^Q (ein((bj ,s)-Yj ) — e-in((bj ,s)-Yj ))(—ab1 )-s1 , , , ( — abm )-sm = j=1

N-m

= n (ein(l'bj S'-Yj ) — e-in{(bj ,s)-Yj ))e-2in(si+...+sm)(ab1 )-si ■ ■ (abm )-sm j=l

is a linear combination of terms (e2in^1 abl )-si ... (e2in^mabm )-sm, where 3 = (¡1;..., ¡m) G zm; the coefficients of this linear combination depend on 3 and 7. Hence <f>(a) is a finite linear combination of shifted integrals as follows

4>(a) = E^(y)i^- f ft+ (bj,s))(z1)-si ... (z-)-Smds,

p (2ni) J5+iRm j

where zj = e2ni^j ab and for a uniform notation, we henceforth denote all rows of the matrix

B = (B',Em)tr by bj.

Therefore the hypergeometric series (8) and (9) can be rewritten as a linear combination of shifted integrals

Ip(z)= I(e2^1 zi,.. ., e2niPmzm),

where

1 r N

1 (z) = ^— II r«bj,s)- Yj)z-sds. (18)

N

Recall that by the condition on the matrix B we have that bj =0 and that means that (18)

1

is a non-confluent Mellin-Barnes integral [17]. The integral (18) tells us how to define a continuous analogue of the T-series.

5. r-integrals

5.1. Definition of r-integrals and their hypergeometricity

Notice that instead of (1) we could consider a series where all T-factors are in the numerator, making use of the fact that the formula r(t)r(1 — t) = n/ sin nt enables us to move terms between numerator and denominator. Using the notations

NN

r(k — 7) = IIr(kj — Yj), aY-k = n a]-kj,

1 1

we could consider a formal version of the T-series on the form

Er(fc - 7)

vY-k,

keL

which also satisfies the hypergeometric system (5).

Let LC = L ® c be the subspace KerA c cn defined by the linear operator A : cn ^ cn. Here A is the integer matrix as before.

Definition 10. We call the integral

$(a) = $7(a) = i r(£ - y)a7_dp (19)

JhcL c

a r-integral, or an A-hypergeometric integral (of the Mellin-Barnes type). Here h is a cycle of dimension m = N — n with a closed support and dp is an m-dimensional measure on LC.

Usually we take dp as the differential form

—— ds = ---— ds-i A ... A dsm,

' \ 'ryi 1 ' \ 'ryi -1- I I V /

(2ni)m (2ni)r

where s G cm is a parameter on L = {£ = Bs} and h as a vertical subspace a + ir c cm. So we get the integral

N

(2ni)~

J — -1-

coinsiding up to a factor aY with the integral (18) after substitution z = aBs.

= ûàm / ft ^>")- 7j)a'y-Bsds (20)

(2ni) Ja+iRm j=1

Proposition 5. Under assumption that the integration set h is homologically equivalent in the holomorphy domain of r(£ — y)|Lc to all shifts h + k, where k runs over the basis of L, the

N

T-integral (19) satisfies the hypergeometric system (5) with 3 = ^2 7ja(j).

1

Proof. According to the Remark 1 after Definition 2 we choose any basis vector k G L and write it as the sum k = k+ — k- for non-negative vectors k+ and k-. Then

N k±j

aY- = ( —l)|k±l H № — 7j + l)aY-^-k±,

d\k±\

j=i1=1

Using the formula tr(t) = r(t + 1) this equality for k_ yields the following _= ( — 1)IM Jh r(e — Y + k-)a?_^_k-dp.

The same is true for k+ = k + k_, hence under the assumption h ~ h + k, one gets

dIk+I

dak+

$7 = ( —1)|k+l i r(C — 7 + k + k-)a?-i-k-k- dp = Jh

( —1)|k+l i r(e — 7 + k-)a?-^-k-h

dp.

Now we recall that |k+| = |k_| for k G L. Hence Dk$7 = 0 for all basis vectors k G L. The equations (Ei — 3i)$ = 0 in (5) are satisfied by integral (19), since the action & — 3i on

N CI

the integrand gives a factor ^ ai'C that vanishes on h c LC. □

3=1

In the case when the T-integral has the form (20), Beukers [20] has given a more effective condition for hypergeometricity.

Proposition 6 ([20], Theorem 3.1). Assume that in integral (3), S = 0 and j & , i.e. ji < 0 for all coordinates ji of j. Then this integral satisfies the A-hypergeometric system (5) with

N

3 = £ Yj 1

For the proof it is enough to show that in the expressions above for ^ k one has

d\k+\ dak+

i r(Bs - y + k-)aY-Bs-k-ds = i r(Bs - y + k-)aY-Bs-k-ds, J s+i Rm Ji Rm

where s satisfies the equation Bs = k. It follows from the fact that the family of subspaces

ts + ir—, t € [0,1] gives a homotopy of cycles s + ir— and ir— in the holomorphy domain of

the integrand, since we we have the for y € r—

B(ts) - y + k- = tk - y + k- = tk+ + (1 - t)k- - y € r—.

5.2. Domain of convergence for the T-integral

Recall also that a Minkowski sum of line segments is called a zonotope (see, for example [21]).

Theorem 4. The convergence domain of the integral (18) is equal to Arg-1(ZB) with ZB being the interior of the zonotope ZB = [0, nb{] + ... + [0, nbN].

Proof. By Theorem 3 one has to prove that ZB = P where

n

P = {0 € r— : \(v,y)\ < 2 g(vv), ^ =1,...,d}.

Since N b- = 0 any point 0 = n(Xb1 + ... + XN bN) € ZB, where X- € [0,1], can also be represented as

0 = -n((1 - X1)b1 + ... + (1 - xn )bN).

This implies that

zb = nb1,nb1] + ... + [-nbN ,nbN ]). Hence any point 0 € ZB can be represented as

0 = 2 • n(Xib1 + ... + Xn bN), with X- € [-1,1]. (21)

By the Triangle Inequality we then get for 0 € ZB

\{e,y)\< lb>v)\fora11 y e Rm>

which means that ZB c PB.

Now we prove that in fact ZB = P. The vector vv in the definition of PB is orthogonal to some subset Bv c B of n — 1 linearly independent vectors bj. Consider the parallelepiped Pv generated by these vectors,

Pv = = n E xjbj : xj e [—1,1]}-

j:bj eBv

The zonotope ZB has two parallel facets being the two extremal translations of P v, namely

F± = Pv + 2 E ^

i:bi€B\Bv

where A± = sign(vv, bj). The normal vector to F± is vv and by (21) for each 9 G ZB

j

n 2

(vv,9) = - J2 X± Vvb).

.:bieB\B

We see that

N

\(v v,9)\ = l(v vj ,bj )\-

j=1

Since ZB = , where Sv is the strip

(vv, 91 ^ - > {Vv,bj) = -g(v

N

Sv = {9;\(vv,9)\< ^(vv,bj) = ng(vv)} j=i

we get our statement ZB = PB. □

5.3. Independence of T-integrals

Now, if we choose coordinates s = (si,..., sm) G LC and represent each £ e LC as a linear combination of column vectors bj of the matrix B,

£ = sib1 + ... + smbm,

then we reduce the T-integral (19) into an integral of type (18) with z = ab. Hence the integral (18) satisfies the hypergeometric system. Since the domain of convergence of the shifted integrals Ip(z) are shifted zonotopes Z@ = —¡3 + ZB, any collection of shifted integrals with non-empty common domain of convergence is linearly independent. More precisely we have the following lemma:

Lemma 1. Let {Zpbe a family of pairwise different shifts of the zonotope ZB. If their intersection

ZJ := p| Zp

peJ

has a non-empty interior, then the corresponding shifts Ip(z) of the integral (3) is linearly independent.

Proof. Let ZPo, 3o G J, be an arbitrary zonotope from the chosen family. Suppose that it contributes to the boundary of the intersection ZJ, that is, there exists a point 90 G dZPo, which is an interior point of the intersection p|peJ\p0 Zp. Then we claim that in any linear relation between the integrals {Ip}peJ the coefficient of Ip0 must be zero. Indeed, otherwise from the representation

IPo(z) = E cpIp(z)

peJ\Po

we would get from Theorem 4 that Ip0 is holomorphic in the sectorial domain over

( n Zp

peJ\po

which contains points from rn \ ZPo. Then it would follow from Theorem 4 and Bochner's theorem that Ip0 would be holomorphic in a sectorial domain over a bigger convex set than Zp0. But this would contradict Theorem 4. From this argument we conclude that, if there exists a linear relation between the Ip, not involving Ip0, then Zp0 contains the intersection of all the other Zp, 3 G J \ /30. Applying the same reasoning to this latter family, we arrive at a situation where all the zonotopes in the family contribute to the boundary of the intersection, and for which the corresponding family of integrals is linearly independent. □

Finally we will need two results from [22] that we for completeness list below. The following theorem reveals a close connection between the discriminant coamoeba and the zonotope.

Theorem 5. The summed chain A'B + ZB is a cycle, and hence equal to mB T2, with some integer multiplicity mB. In fact, provided that the vectors bk are ordered clockwise projectively, this multiplicity is given by the formula

mB = 2 Edet+(bj, bk).

j<k

Theorem 6. The multiplicity mB from Theorem 5 coincides with dB, where dB is the normalized volume of the convex hull of the point configuration A c Z2.

Using these results of [22], we arrive at the following result in two dimensions. We use the notation m for the normalized area of conv(A) which is equal to the maximal number of linearly independent solutions to the hypergeometric system of differential equations (5).

Theorem 7. Assume n = 2. Let Fi be a component of the torus (r/2nz)2 that is covered by the coamoeba A'Ea exactly i times. Then there will be exactly m — i integrals of the type (18) converging in the sectorial domain over Fi. In particular for the complement of the coamoeba, that is for i = 0, these integrals provide a basis for the whole solution space to (5).

Proof. Follows from Lemma 1, Theorem 5 and Theorem 6. □

The third author was supported by the grant of Ministry of Education and Science of the Russian Federation (no. 1.2604.2017/PCh) and was supported by RFBR, grant 18-51-41011 Uzb.t.

References

[1] I.Gelfand, M.Kapranov, A.Zelevinsky, Hypergeometric functions and toric varieties, Funct. Anal. Appl., 23(1989), no. 2, 94-106

[2] J.Horn, Uber die Convergenz der hypergeometrischen Reihen zweier und dreier Veranderlichen, Math. Ann., 34(1889), 544-600.

[3] B.Grunbaum, Convex polytopes, Graduate Texts in Mathematics 221, Springer Verlag, 2nd ed. 2003.

[4] A.Tsikh, O.Zhdanov, Investigation of multiple Mellin-Barnes integrals by means of multidimensional residues, Siberian Math. J., 39(1998), no. 2, 245-260.

[5] L.Nilsson, Amoebas, Discriminants, and Hypergeometric Functions, Doctoral thesis in mathematics at Stocholm University, Sweden, 2009.

[6] H.Mellin, Mellin Resolution de l'equation algebrique generale a l'aide de la fonction gamma, C. R. Acad. Sci. Paris Ser. I Math., 172(1921), 658-661.

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Области сходимости А-гипергеометрических рядов и интегралов

Лиса Нильсон Микаэль Пассаре

Стокгольмский университет Стокгольм Швеция

Август К. Цих

Институт математики и фундаментальной информатики Сибирский федеральный университет Свободный, 79, Красноярск, 660041

Россия

Доказываются две теоремы об областях сходимости для А-гипергеометрических рядов и ассоциированных с ними интегралов типа Меллина-Барнса. Точные области сходимости описаны в терминах амеб и коамеб соответствующих главных А-детерминантов.

Ключевые слова: А-гипергеометрический ряд, интеграл Меллина-Барнса, Г-интеграл, главный А-дискриминант.