An ansatz for the asymptotics of hypergeometric multisums

An ansatz for the asymptotics of hypergeometric multisums

Stavros Garoufalidis School of Mathematics
Georgia Institute of Technology
Atlanta, GA 30332-0160, USA
http://www.math.gatech
.edu/stavros
stavros@math.gatech.edu
February 14, 2008
Abstract.

Sequences that are defined by multisums of hypergeometric terms with compact support occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory. The standard recipe to study the asymptotic expansion of such sequences is to find a recurrence satisfied by them, convert it into a differential equation satisfied by their generating series, and analyze the singulatiries in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set, for which we conjecture (and in some cases prove) that it contains all the singularities of the generating series. Our construction of this finite set is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. The finite set can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive -theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.

The author was supported in part by NSF.

1991 Mathematics Classification. Primary 57N10. Secondary 57M25.
Key words and phrases: holonomic functions, regular holonomic functions, special hypergeometric terms, WZ algorithm, diagonals of rational functions, asymptotic expansions, transseries, Laplace transform, Hadamard product, Bethe ansatz, singularities, additive -theory, regulators, Stirling formula, algebraic combinatorics, polytopes, Newton polytopes, -functions.

1. Introduction

1.1. The problem

The problem considered here is the following: given a balanced hypergeometric term with compact support for each , we wish to find an asymptotic expansion of the sequence

(1)

Such sequences occur frequently in enumeration problems of combinatorics, algebraic geometry and perturbative quantum field theory; see [St, FS, KM]. The standard recipe for this is to find a recurrence satisfied by , convert it into a differential equation satisfied by the generating series

and analyze the singulatiries of its analytic continuation in the complex plane. We propose a shortcut by constructing directly from the structure of the hypergeometric term a finite set , for which we conjecture (and in some cases prove) that it contains all the singularities of . Our construction of is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. can also be identified with the set of critical values of a potential function, as well as with the evaluation of elements of an additive -theory group by a regulator function. We give a proof of our conjecture in some special cases, and we illustrate our results with numerous examples.

1.2. Existence of asymptotic expansions

A general existence theorem for asymptotic expansions of sequences discussed above was recently given in [Ga4]. To phrase it, we need to recall what is a -function in the sense of Siegel; [Si].

Definition 1.1.

We say that series is a -function if

  • the coefficients are algebraic numbers, and

  • there exists a constant so that for every the absolute value of every conjugate of is less than or equal to , and

  • the common denominator of is less than or equal to .

  • is holonomic, i.e., it satisfies a linear differential equation with coefficients polynomials in .

The main result of [Ga4] is the following theorem.

Theorem 1.

[Ga4, Thm.3] For every balanced term , the generating series is a -function.

Using the fact that the local monodromy of a -function around a singular point is quasi-unipotent (see [Ka, An2, CC]), an elementary application of Cauchy’s theorem implies the following corollary; see [Ga4], [Ju] and [CG1, Sec.7].

Corollary 1.2.

If is a -function, then has a transseries expansion, that is an expansion of the form

(2)

where is the set of singularities of , , , and . In addition, is a finite set of algebraic numbers, and generates a number field .

1.3. Computation of asymptotic expansions

Theorem 1 and its Corollary 1.2 are not constructive. The usual way for computing the asymptotic expansion for sequences of the form (1) is to find a linear reccurence, and convert it into a differential equation for the generating series . The singularities of are easily located from the roots of coefficient of the leading derivative of the ODE. This approach is taken by Wimp-Zeilberger, following Birkhoff-Trjitzinsky; see [BT, WZ2] and also [Ni]. A reccurence for a multisum sequence follows from Wilf-Zeilberger’s constructive theorem, and its computer implementation; see [Z, WZ1, PWZ, PR1, PR2]. Although constructive, these algorithms are impractical for multisums with, say, more than three summation variables.

On the other hand, it seems wasteful to compute an ODE for , and then discard all but a small part of it in order to determine the singularities of .

The main result of the paper is a construction of a finite set of algebraic numbers directly from the summand , which we conjecture that it includes the set . We give a proof of our conjecture in some special cases, as well as supporting examples.

Our definition of the set is reminiscent of the Bethe ansatz, and is related to critical values of potential functions and additive -theory.

Before we formulate our conjecture let us give an instructive example.

Example 1.3.

Consider the Apery sequence defined by

(3)

It turns out that satisfies a linear recursion relation with coefficients in (see [WZ2, p.174])

(4)

for all , with initial conditions . It follows that is holonomic; i.e., it satisfies a linear ODE with coefficients in :

(5)

with initial conditions , , .

This implies that the possible singularities of are the roots of the equation:

(6)

Thus, has analytic continuation as a multivalued function in . Since the Taylor series coefficients of at are positive integers, and is analytic at , it follows that has a singularity inside the punctured unit disk. Thus, is singular at . By Galois invariance, it is also singular at . The proof of Corollary 1.2 implies that has an asymptotic expansion of the form:

(7)

for some constants with . A final calculation shows that

Our paper gives an ansatz that quickly produces the numbers given the expression (3) of , bypassing Equations (4) and (5). This is explained in Section 5.2. In a forthcoming publication we will explain how to compute the Stokes constants for , in terms of the expression (3).

1.4. Hypergeometric terms

We have already mentioned multisums of balanced hypergeometric terms. Let us define what those are.

Definition 1.4.

An -dimensional balanced hypergeometric term (in short, balanced term, also denoted by ) in variables , where and , is an expression of the form:

(8)

where are algebraic numbers for , for , and are integral linear forms in that satisfy the balance condition:

(9)
Remark 1.5.

An alternative way of encoding a balanced term is to record the vector , and the matrix of the coefficients of the linear forms , and the signs for .

1.5. Balanced terms, generating series and singularities

Given a balanced term , we will assign a sequence and a corresponding generating series to a balanced term , and will study the set of singularities of the analytic continuation of in the complex plane.

Let us introduce some useful notation. Given a linear form in variables where , and , let us define

(10)

For we define:

(11)
Definition 1.6.

Given a balanced term  as in (8), define its Newton polytope by:

(12)

We will assume that is a compact rational convex polytope in with non-empty interior. It follows that for every we have:

(13)
Definition 1.7.

Given a balanced term  consider the sequence:

(14)

(the sum is finite for every ) and the corresponding generating function:

(15)

Here denote the field of algebraic numbers. Let denote the finite set of singularities of , and denote the corresponding number field, following Corollary 1.2.

Remark 1.8.

Notice that determines but not vice-versa. Indeed, there are nontrivial identities among multisums of balanced terms. Knowing a complete set of such identities would be very useful in constructing invariants of knotted objects, as well as in understanding relations among periods; see [KZ].

Remark 1.9.

The balance condition of Equation (9) is imposed so that for every balanced term  the corresponding sequence grows at most exponentially. This follows from Stirling’s formula (see Corollary 2.1) and it implies that the power series is the germ of an analytic function at . Given a proper hypergeometric term in the sense of [WZ1], we can find and so that is a balanced term.

1.6. The definition of and

Let us observe that if is a balanced term and is a face of its Newton polytope , then is also a balanced term.

Definition 1.10.

Given a balanced term  as in Equation (8) consider the following system of Variational Equations:

(16)

in the variables . Let denote the set of complex solutions of (16), with the convention that when we set , and define

(17)
(18)
(19)
Remark 1.11.

There are two different incarnations of the set : it coincides with

  • the set of critical values of a potential function; see Theorem 5.

  • the evaluation of elements of an additive -theory group under the entropy regulator function; see Theorem 6.

It is unknown to the author whether is always a finite set. Nevertheless, is always a finite subset of ; see Theorem 5. Equations (16) are reminiscent of the Bethe ansatz.

1.7. The conjecture

Section 1.5 constructs a map:

via generating series and their singularities, where is a number field and is a finite subset of . Section 1.6 constructs a map:

via solutions of polynomial equations. We are now ready to formulate our main conjecture.

Conjecture 1.

For every balanced term  we have: and consequently, .

1.8. Partial results

Conjecture 1 is known to hold in the following cases:

  • For -dimensional balanced terms: see Theorem 2.

  • For positive special terms: see Theorem 3.

  • For -dimensional balanced terms, see [Ga3].

Since the finite sets and that appear in Conjecture 1 are in principle computable (as explained in Section 1.3), one may try to check random examples. We give some evidence in Section 5. We refer the reader to [GV] for an interesting class of -dimensional examples related to -symbols, and of interest to atomic physics and low dimensional topology.

The following proposition follows from the classical fact concerning singularities of hypergeometric series; see for example [O, Sec.5] and [No].

Theorem 2.

Suppose that is 0-dimensional balanced term as in (8), with . Then is a hypergeometric series and Conjecture 1 holds.

When is positive dimensional, the generating series is no longer hypergeometric in general. To state our next result, recall that a finite sybset of algebraic numbers is irreducible over if the Galois group acrs transitively on .

Definition 1.12.

A special hypergeometric term (in short, special term) is an expression of the form:

(20)

where , and and are integral linear forms in . We will assume that for every , the support of as a function of is finite. We will call such a term positive if for .

Lemma 1.13.

(a) A balanced term is the ratio of two special terms. In other words, it can always be written in the form:

(21)

for some integral linear forms and signs .
(b) The set of special terms is an abelian monoid with respect to multiplication, whose corresponding abelian group is the set of balanced terms.

The proof of (a) follows from writing a balanced term in the form:

where

To illustrate part (a) of the above lemma for -dimensional balanced terms, we have:

The above identity also shows that if a balanced term takes integer values, it need not be a special term. This phenomenon was studied by Rodriguez-Villegas; see [R-V].

Theorem 3.

Fix a positive special term  such that is irreducible over . Then, and Conjecture 1 holds.

In a forthcoming publication we will give a proof of Conjecture 1 for 1-dimensional balanced terms; see [Ga3]. Let us end this section with an inverse type (or geometric realization) problem.

Problem 1.14.

Given , does there exist a special term so that ?

1.9. Laurent polynomials: a source of special terms

This section, which is of independent interest, associates an special term  to a Laurent polynomial with the property that the generating series is identified with the trace of the resolvant of . Combined with Theorem 1, this implies that is a -function.

If is a square matrix of size with entries Laurent polynomials in variables, let denote the constant term of its usual trace. The moment generating series of is the power series

(22)
Theorem 4.

(a) For every there exists a special term  so that

(23)

Consequently, is a -function.
(b) The Newton polytope depends is a combinatorial simplex which depends only on the monomials that appear in and not on their coefficients.
(c) For every , is a -function.

We thank C. Sabbah for providing an independent proof of part (c) when using the regularity of the Gauss-Manin connection. Compare also with [DvK].

1.10. Plan of the proof

In Section 2, we introduce a potential function associated to a balanced term  and we show that the set of its critical values coincides with the set that features in Conjecture 1. This also implies that is finite.

In Section 3 we assign elements of an extended additive -theory group to a balanced term , and we show that the set of their values (under the entropy regulator map) coincides with the set that features in Conjecture 1.

In Section 4 we give a proof of Theorems 3 (using results from hypergeometric functions) and 2 (using an application of Laplace’s method), which are partial case of our Conjecture 1.

In Section 5 we give several examples that illustrate Conjecture 1.

In Section 6 we study a special case of Conjecture 1, with input a Laurent polynomial in many commuting variables.

1.11. Acknowledgement

The author wishes to thank Y. André, N. Katz, M. Kontsevich, J. Pommersheim, C. Sabbah, and especially D. Zeilberger for many enlightening conversations, R.I. van der Veen for a careful reading of the manuscript, and the anonymous referee for comments that improved the exposition. An early version of the paper was presented in an Experimental Mathematics Seminar in Rutgers in the spring of 2007. The author wishes to thank D. Zeilberger for his hospitality.

2. Balanced terms and potential functions

2.1. The Stirling formula and potential functions

As a motivation of a potential function associated to a balanced term , recall Stirling formula, which computes the asymptotic expansion of (see [O]):

(24)

For , we can define . The Stirling formula implies that for fixed and large, we have:

(25)

The next corollary motivates our definition of the potential function.

Corollary 2.1.

For every balanced term  as in (8) and every in the interior of we have:

(26)

where the potential function is defined below.

Definition 2.2.

Given a balanced term  as in (8) define its corresponding potential function by:

(27)

where ,

(28)

and

(29)

is a multivalued analytic function on the complement of the linear hyperplane arrangement , where

(30)

In fact, let

(31)

denote the universal abelian cover of .

The next theorem relates the critical points and critical values of with the sets and from Definition 1.10.

Theorem 5.

(a) For every balanced term , coincides with the set of critical points of .
(b) If is a critical point of , then

(32)

Thus coincides with the exponential of the set of the negatives of the critical values of .
(c) For every face of the Newton polytope of we have:

(d) is a finite subset of and is a number field.

2.2. Proof of Theorem 5

Let us fix a balanced term  as in (8) and a face of its Newton polytope . Without loss of generality, assume that . Since

(33)

it follows that for every and every we have:

(34)

Adding up with respect to , using the balancing condition of Equation (9), and the notation of Section 1.5 applied to the linear form of Equation (28), it follows that

This proves that the critical points of are the solutions to the following system of Logarithmic Variational Equations:

(35)

where, for a subgroup of and an integer , we define

(36)

Exponentiating, it follows that satisfies the Variational Equations (16), and concludes the proof of part (a).

For part (b), we will show that if is a critical point of , the corresponding critical value is given by:

(37)

Exponentiating, we deduce the first equality of Equation (32). The second equality follows from the fact that for all critical points of .

To show (37), observe that for any linear form we have:

Suppose that satisfies the Logarithmic Variational Equations (35). Using the definition of the potential function, and collecting terms with respect to it follows that

This concludes part (b). Part (c) follows from set-theoretic considerations, and part (d) follows from the following facts:

  • an analytic function is constant on each component of its set of critical points,

  • the set of critical points are the complex points of an affine variety defined over by (16),

  • every affine variety has finitely many connected components.

This concludes the proof of Theorem 5. ∎

3. Balanced terms, the entropy function and additive -theory

In this section we will assign elements of an extended additive -theory group to a balanced term, and using them, we will identify our finite set from Definition (1.10) with the values of the constructed elements under a regulator map; see Theorem 6.

3.1. A brief review of the entropy function and additive -theory

In this section we will give a brief summary of an extended version of additive -theory and the entropy function following [Ga2], and motivated by [Ga1]. This section is independent of the rest of the paper, and may be skipped at first reading.

Definition 3.1.

Consider the entropy function , defined by:

(38)

for .

is a multivalued analytic function on , given by the double integral of a rational function as follows from:

(39)

For a detailed description of the analytic continuation of , we refer the reader to [Ga2]. Let denote the universal abelian cover of . In [Ga2, Sec.1.3] we show that has an analytic continuation:

(40)

In [Ga2, Def.1.7] we show that satisfies three 4-term relations, one of which is the analytic continuation of (48). The other two are dictated by the variation of along the cuts and .

Using the three 4-term relations of , we introduce an extended version in [Ga2, Def.1.7]:

Definition 3.2.

The extended group is the -vector space generated by the symbols with , subject to the extended 4-term relation:

(41)

for , and the relations:

(42)
(43)

for , .

Since the three 4-term relations in the definition of are satisfied by the entropy function, it follows that gives rise to a regulator map:

(44)

For a motivation of the extended group and its relation to additive (i.e., infinitesimal) -theory and infinitesimal polylogarithms, see [Ga2, Sec.1.1] and references therein.

3.2. Balanced terms and additive -theory

In this section, it will be more convenient to use the presentation (21) of balanced terms. In this case, we have:

(45)

The Stirling formula motivates the constructions in this section. Indeed, we have the following:

Lemma 3.3.

For , we have:

(46)

and

(47)
Proof.

Equation (46) is elementary. Equation (47) follows from the Stirling formula (24). ∎

The next lemma gives a combinatorial proof of the 4-term relation of the entropy function.

Lemma 3.4.

For , satisfies the 4-term relation:

(48)
Proof.

The 4-term relation follows from the associativity of the multinomial coefficients

(49)

applied to and Lemma 3.3, and the specialization to . In fact, the 4-term relation (48) and a local integrability assumption uniquely determines up to multiplication by a complex number. See for example, [Da] and [AD, Sec.5.4,p.66]. ∎

Corollary 3.5.

If is a balanced term as in (21), then its potential function is given by:

(50)

Consider the complement of the linear hyperplane arrangement given by:

(51)

Let

(52)

denote the universal abelian cover of . For , the functions , have analytic continuation:

It follows that the potential function given by (50) has an analytic continuation:

(53)

Let denote the set of critical points of .

Definition 3.6.

For a balanced term  as in (21), we define the map:

(54)

by:

Theorem 6.

For every balanced term , we have a commutative diagram:

3.3. Proof of Theorem 6

We begin by observing that the analytic continuation of the logarithm function gives an analytic function:

The proof of part (a) of Theorem 5 implies that if and only if satisfies the Logarithmic Variational Equations:

(55)

for . Exponentiating, this implies that .

The proof of part (b) of Theorem 5 implies that if , then:

On the other hand, if , we have:

where the last equality follows from the analytic continuation of (46). Expanding the linear forms , , and with respect to the variables for , and using the Logarithmic Variational Equations (55), (as in the proof of part (b) of Theorem 5), it follows that

Thus,

This concludes the proof of Theorem 6.

4. Proof of Theorems 2 and 3

4.1. Proof of Theorem 2

A -dimensional balanced term is of the form:

where , for satisfying . Since , it follows that

is so-called closed form. The Newton polytope of is given by . In addition, and for . By definition, we have , and

(56)

On the other hand, is a hypergeometric series with singularities ; see for example [O, Sec.5] and [No]. The result follows.

4.2. Proof of Theorem 3

The proof of Theorem 3 is a variant of Laplace’s method and uses the positivity of the restriction of the potential function to . See also [Kn, Sec.5.1.4]. Suppose that for all . Recall the corresponding polytope and consider the restriction of the potential function to :

It is easy to show that for . This is illustrated by the plot of the entropy function for :

It follows that the restriction of the potential function on is nonnegative and continuous. By compactness it follows that the function achieves a maximum in in the interior of . It follows that for every we have: