key: cord-0047205-xi8npk57
authors: Matsubara-Heo, Saiei-Jaeyeong; Takayama, Nobuki
title: Algorithms for Pfaffian Systems and Cohomology Intersection Numbers of Hypergeometric Integrals
date: 2020-06-06
journal: Mathematical Software - ICMS 2020
DOI: 10.1007/978-3-030-52200-1_7
sha: bfb1e33be2c1f2dbe87c8030e81b56298d39a714
doc_id: 47205
cord_uid: xi8npk57

In the theory of special functions, a particular kind of multidimensional integral appears frequently. It is called the Euler integral. In order to understand the topological nature of the integral, twisted de Rham cohomology theory plays an important role. We propose an algorithm of computing an invariant cohomology intersection number of a given basis of the twisted cohomology group. We also develop an algorithm of computing the Paffian system that a given basis satisfies. These algorithms are based on the fact that the Euler integral satisfies GKZ system and utilizes algorithms to find rational function solutions of differential equations. We provide software to perform this algorithm.

In the study of hypergeometric functions in several variables, one often considers the integral of the following form:

where

j x a (l) (j) (l = 1, . . . , k) are Laurent polynomials in torus variables x = (x 1 , . . . , x n ), a (l) (j) ∈ Z n , γ l ∈ C and c = t (c 1 , . . . , c n ) ∈ C n are parameters, x c = x c1 1 . . . x cn n , Γ is a suitable integration cycle, and ω is an algebraic n-form on V z = {x ∈ C n | x 1 . . . x n h 1 (x) . . . h k (x) = 0}. As a function of the independent variable z = (z (l) j ) j,l , the integral (1) defines a hypergeometric function. We call the integral (1) the Euler integral.

We can naturally define the twisted de Rham cohomology group associated to the Euler integral (1) . We set N = N 1 +· · ·+N k , G n m = Specm C[x ± 1 , . . . , x ± n ], and A N = Specm C[z (l) j ]. For any z ∈ A N , we can define an integrable connection

Vz . The algebraic de Rham cohomology group H * dR (V z ; (O Vz , ∇ x )) is defined as the hypercohomology group

Under a genericity assumption on the parameters γ l and c, we have the vanishing result H m dR (V z ; (O Vz , ∇ x )) = 0 (m = n). Moreover, we can define a perfect pairing •,

The study of intersection numbers of twisted cohomology groups and twisted period relations for hypergeometric functions started with the celebrated work by K. Cho and K. Matsumoto [6] . They clarified that the cohomology intersection number appears naturally as a part of the quadratic relation, a class of functional relations of hypergeometric functions. They also developed a systematic method of computing the cohomology intersection number for 1-dimensional integrals. Since this work, several methods have been proposed to evaluate intersection numbers of twisted cohomology groups, see, e.g., [2, 3, 10, 11, 14, 17, 19] and references therein. All methods utilize comparison theorems of twisted cohomology groups and residue calculus.

When z belongs to a certain non-empty Zariski open subset of A N (the non-singular locus), we proposed a new method in the paper [16] to obtain cohomology intersection numbers by constructing a rational function solution of a system of linear partial differential equations. One weak point of the method was that it was not algorithmic to construct the Pfaffian system (the explicit form of the integrable connection) for a given basis of the twisted cohomology group. We will give a new algorithm to construct the Pfaffian system for a given basis in this paper (Algorithm 1). To our knowledge, algorithms to find the Pfaffian system (or equation) with respect to a given basis of twisted cohomology group do not appear in the literature except the twisted logarithmic cohomology case 1 . Our algorithm works for a more general class of twisted cohomology groups. Moreover, it is more efficient by utilizing Saito's b-function [23] expressed in terms of facets of a polytope. The Sect. 2 is a brief overview of the paper [16] . The Sect. 3 is the main part and in the Sects. 4 and 5, we will give demonstrations of our implementation. As to the construction of rational function solutions, we utilize the algorithm and the implementation by M. Barkatou, T. Cluzeau, C. El Bacha, J.-A. Weil [5] (see also [4, 18] and their references).

, ∇ an x ) . Note that the Poincaré dual of the isomorphism reg is called a regularization map in the theory of special functions ([2, § 3.2]). Finally, we define the cohomology intersection form •, • ch between algebraic de Rham cohomology groups by the formula

The value above is called the cohomology intersection number of φ and ψ.

Now, we treat z as a variable. Let π : 

Here, Ω • X/Y denotes the sheaf of relative differential forms ⊕ |I|=• O X dx I with respect to the morphism π. Since Y is affine, H n dR is also identified with the sheaf R n π 

Here, the superscript GM stands for "Gauß-Manin". The dual connection 

We call (7) the secondary equation. Let us rewrite it in terms of local frames.

Then, the secondary equation is equivalent to the system

We also call (8) the secondary equation. The theorem which our algorithm is based on is the following Theorem 1 [16] . Under the regularization condition, all the entries of the cohomology intersection matrix I ch are rational functions. Moreover, any rational function solution I of the secondary equation (8) is, up to a scalar multiplication, equal to I ch .

In [16] , it is discussed that Theorem 1 is true for more general direct image D-modules. However, by employing the combinatorial structure behind our integrable connection H n dR U , we can show that the cohomology intersection number in question has a rational expression with respect to z and δ.

Let us recall the definition of GKZ system [8] . For a given d × N (d < N) integer matrix A = (a(1)| · · · |a(N )) and a parameter vector δ ∈ C d , GKZ system M A (δ) is defined as a system of partial differential equations on C N given by

where E i and u for u = t (u 1 , . . . , u N ) are differential operators defined by

For convenience, we assume an additional condition ZA

We put δ = γ c . By abuse of notation, we also denote by M A (δ) the quotient

) and the regularization condition is true when the parameter vector δ is non-resonant and γ l / ∈ Z (see [9, 2.9 ] and [15, Theorem 2.12]). We set dx

Thus, any section φ of H n dR can be written as φ = P · [ dx x ] for some linear differential operator P ∈ D Y . We define the field Q(δ) as the field extension Q(γ 1 , . . . , γ k , c 1 , . . . , c n ) of Q.

Theorem 2 [16] . Suppose that A as in (11) admits a unimodular regular triangulation T and δ is non-resonant and γ l / ∈ Z. Then, for any P 1 , P 2 ∈ Q(δ) z, ∂ z , the cohomology intersection number

belongs to the field Q(δ)(z).

In this section, we set β := −δ. With this notation, we put H A (β) := M A (δ). This is because we use some results from [12] and [23] where the hypergeometric ideal is denoted by H A (β) while our main references [15, 16] denote it by M A (δ).

Let ω q be the differential form

It is known that there exists a basis of the twisted cohomology group of which elements are of the form ω q when δ is generic (see, e.g., [13, Th 2] ). Let {ω q | q ∈ Q} be a basis of the twisted cohomology group. We will give an algorithm to find the Pfaffian system ∂ ∂zi ω = P i ω with respect to this basis ω = (ω q | q ∈ Q) T . Note that algorithms to translate a given holonomic ideal to a Pfaffian system are well known (see, e.g., [12, Chap 6] ). However, as long as we know, algorithms to find the Pfaffian system with respect to a given basis of twisted cohomology group do not appear in the literature. Note that the pairing of the twisted homology and cohomology groups is perfect under our assumption. Then, the Pfaffian equation of the fundamental solution matrix of solutions of the GKZ system can be regarded as a relation of the twisted cycles.

Put ∂ i = ∂ ∂zi . In this subsection, we use • to denote the action to avoid a confusion with the multiplication. The function ω q is a solution of the hypergeometric system H A (β − q). The main point of our method is of use of the following contiguity relation 

In [23, Algorithm 3.2], an algorithm to obtain an operator C i satisfying

is given. The polynomial b i is a b-function in the direction i [23, Th 3.2]. Note that the algorithm outputs the operator C i in C z 1 , . . . , z N , ∂ 1 , . . . , ∂ N , which does not depend on the parameter β. Since ω q is a solution of H A (β − q), we have the following inverse contiguity relation 

Then, h 1 = z 1 + z 2 x, h 2 = z 3 + z 4 x. We have

We can show that {ω (1, 0, 0) , ω (1,0,0) − ω (0,1,0) } is a basis of the twisted cohomolgy group. This A is normal and the b-function b 4 (s) ∈ Q[s 1 , s 2 , s 3 ] for the direction

Our algorithm to find a Pfaffian system with respect to a given basis of the twisted cohomology group is as follows. Algorithm 1. Input: {ω q | q ∈ Q}, a basis of the twisted cohomology group. A direction (index) i. Output: P i , the coefficient matrix of the Pfaffian system ∂ i − P i .

with rational function coefficients. Let S be a column vector of the standard monomials with respect to G.

It is a vector with entries in the ring of differential operators and the order of the product is i = N, N − 1, . . . , 3, 2, 1 . In other words, we apply operators from ∂ 1 . The polynomial B is derived from the coefficient of the contiguity relation (15) and is equal to

The polynomial B comes from the denominator of the contiguity relation (13) and is equal to

3. Compute the normal form of the vectors ∂ i F (Q) and F (Q). Write the normal forms of them as P S and P S respectively where P and P are matrices with rational function entries. 4. Output P i = P (P ) −1 .

The matrix P is invertible if and only if the given set of differential forms {ω q } is a basis of the twisted cohomology group.

We show the correctness of the algorithm. Take an element q ∈ Q. We express ω q in terms of ω 0 , which is a solution of H A (β), by the contiguity relations (13) and (15) . Note that the contiguity relations for functions ω q give the contiguity relations for cohomology classes [ω q ] by virtue of the perfectness of the pairing between the twisted homology and the twisted cohomology groups. The point of the correctness is the following identity

Let us illustrate how to prove (23) by examples. We assume that q = 2a 1 + a 2 and N 1 ≥ 2. Then ω q can be obtained by applying (13) with i = 1 for two times and that with i = 2. We have

Thus, we obtain the numbers (21) and then (23) . Let us consider the case that q = −2a 1 −a 2 and N 1 ≥ 2. Then ω q can be obtained by applying (15) with i = 1 for two times and that with i = 2. Since ω −a1 is a solution of

from [23] . Then, we have

Thus, we obtain the numbers (19) and then (23) . The general case can be shown by repeating these procedures. When the normal form F (q) with respect to the Gröbner basis G is p i s i where S = (s i ) and p i is a rational function in z and β, we have

The correctness of the last two steps follows from this fact.

Example 2. This is a continuation of Example 1. We have (1, 0, 0) T = a 1 and (0, 1, 0) T = a 3 . Then, the basis of the twisted cohomology group F (Q) is expressed as F (Q) = (∂ 1 /β 1 ,

We can obtain a Gröbner basis whose set of the standard monomials is {∂ 4 , 1} by the graded reverse lexicographic order such that ∂ i > ∂ i+1 . We multiply β 1 β 2 to F (Q) and ∂ 4 F (Q) in order to avoid rational polynomial arithmetic. Then, the normal form, for example, of β 2 ∂ 1 is 1 z1z4−z2z3 (β 1 (β 1 + β 2 )z 4 )∂ 4 − β 2 2 β 3 . By computing the other normal forms, we obtain the matrix

We implemented our algorithms on the computer algebra system Risa/Asir [21] with a Polymake interface. Polymake (see, e.g., [20, 22] ) is a system for polyhedral geometry and it is used for an efficient computation of contiguity relations ([23, Algorithm 3.2]). Here is an input 2 to find the coefficient matrix P 4 for Example 1 with respect to the variable z 4 when z 1 = z 2 = z 3 = 1 (note that in our implementation x is used instead of z). It outputs the following coefficient matrix 

where

When z 2 = −1, z 3 = z 4 = z 5 = z 6 = 1, the coefficient matrix for z 1 for the basis ( ω 1 , ω 2 , ω 3 ) T is

The result can be obtained in a few seconds.

Theorem 3 [16] . Given a matrix A = (a ij ) as in (11) admitting a unimodular regular triangulation T . When parameters are non-resonant, γ l / ∈ Z and moreover the set of series solutions by T is linearly independent, the intersection matrix of the twisted cohomology group of the GKZ system associated to the matrix A can be algorithmically determined.

We denote by Ω i the coefficient matrix of Ω with respect to the 1-form dz i . The algorithm we propose is summarized as follows.

Input: Free bases {φ j } j ⊂ H n dR U , {ψ j } j ⊂ H n∨ dR U which are expressed as (12) .

Output: The secondary equation (8) and the cohomology intersection matrix

1. Obtain a Pfaffian system with respect to the given bases {φ j } j and {ψ j } j , i.e., obtain matrices Ω i = (ω ijk ) and Ω ∨ i = (ω ∨ ijk ) so that the equalities

hold by Algorithm 1.

To be more precise, see, e.g., [4, 5, 18] and references therein. 3. Determine the scalar multiple of I by [15, Theorem 8.1] . Example 4. This is a continuation of Example 3. We want to evaluate the cohomology intersection matrix I ch = ( ω i , ω j ch ) 3 i,j=1 . By solving the secondary equation (for example, using [5] ), we can verify that (1, 1), (1, 2), (2, 1), (2, 2) entries of I ch are all independent of z 1 . Therefore, we can obtain the exact values of these entries by taking a unimodular regular triangulation T = {23456, 12456, 12346} and substituting z 1 = 0 in [15, Theorem 8.1 ]. Thus, we get a correct normalization of I ch and the matrix 

where

Note that this A is not normal. When z 1 = z 4 = z 5 = 1, we have obtained the coefficient matrices P 2 and P 3 in about 9 h 45 min on a machine with Intel(R) Xeon(R) CPU E5-4650 2.70 GHz and 256 GB memory. The (1, 1) element of P 2 is ((b 2 z 2 2 + b 123 )z 2 3 + b 2 z 4 2 + b 132 z 2 2 − 32b 1 + 16b 2 + 16b 3 − 16) z 2 (z 2 − 2)(z 2 + 2)(z 2 3 + z 2 2 − 4)

where b 1 = −γ 1 , b 2 = −c 1 , b 3 = −c 2 and b ijk = 8b i − 4b j − 8b k + 4. A complete data of P 2 and P 3 is at [25] . The intersection matrix can be obtained by [5] in a few seconds when we specialize b i 's to rational numbers. See [25] as to Maple inputs for it.

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