AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS

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1 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS STAVROS GAROUFALIDIS 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 K-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. Contents. Introduction 2.. The problem 2.2. Existence of asymptotic expansions 2.3. Computation of asymptotic expansions 3.4. Hypergeometric terms 4.5. Balanced terms, generating series and singularities 4.6. The definition of S t and K t 5.7. The conjecture 5.8. Partial results 5.9. Laurent polynomials: a source of special terms 7.0. Plan of the proof 7.. Acknowledgement 7 2. Balanced terms and potential functions The Stirling formula and potential functions Proof of Theorem Balanced terms, the entropy function and additive K-theory A brief review of the entropy function and additive K-theory Balanced terms and additive K-theory Proof of Theorem Proof of Theorems 2 and Proof of Theorem Proof of Theorem 3 3 Date: February 4, The author was supported in part by NSF. 99 Mathematics Classification. Primary 57N0. Secondary 57M25. Key words and phrases: holonomic functions, regular holonomic functions, balanced hypergeometric terms, WZ algorithm, diagonals of rational functions, asymptotic expansions, transseries, Laplace transform, Hadamard product, Bethe ansatz, singularities, additive K-theory, regulators, Stirling formula, algebraic combinatorics, polytopes, Newton polytopes, G-functions.

2 2 STAVROS GAROUFALIDIS 5. Some examples A closed form example The Apery sequence An example with critical points at the boundary 5 6. Laurent polynomials and special terms Proof of Theorem The Newton polytope of a Laurent polynomial and its associated balanced term 8 References 2. Introduction.. The problem. The problem considered here is the following: given a balanced hypergeometric term t n,k,...,k r with compact support for each n N, we wish to find an asymptotic expansion of the sequence () a t,n = (k,...,k r) Z r t n,k,...,k r. 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 (a n ), convert it into a differential equation satisfied by the generating series G t (z) = a t,n z n n=0 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 t n,k,...,k r a finite set S t, for which we conjecture (and in some cases prove) that it contains all the singularities of G(z). Our construction of S t is given by the solution set of a balanced system of polynomial equations of a rather special form, reminiscent of the Bethe ansatz. S t 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 K-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..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 G-function in the sense of Siegel; [Si]. Definition.. We say that series G(z) = n=0 a nz n is a G-function if (a) the coefficients a n are algebraic numbers, and (b) there exists a constant C > 0 so that for every n N the absolute value of every conjugate of a n is less than or equal to C n, and (c) the common denominator of a 0,..., a n is less than or equal to C n. (d) G(z) is holonomic, i.e., it satisfies a linear differential equation with coefficients polynomials in z. The main result of [Ga4] is the following theorem. Theorem. [Ga4, Thm.3] For every balanced term t, the generating series G t (z) is a G-function. Using the fact that the local monodromy of a G-function around a singular point is quasi-unipotent (see [Ka, An, An2, CC]), an elementary application of Cauchy s theorem implies the following corollary; see [Ga4], [Ju] and [CG, Sec.7].

3 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 3 Corollary.2. If G(z) = n=0 a nz n is a G-function, then (a n ) has a transseries expansion, that is an expansion of the form (2) a n λ Σ λ n n α λ (log n) β λ where Σ is the set of singularities of G(z), α λ Q, β λ N, and c λ,s C. In addition, Σ is a finite set of algebraic numbers, and generates a number field E = Q(Σ)..3. Computation of asymptotic expansions. Theorem and its Corollary.2 are not constructive. The usual way for computing the asymptotic expansion for sequences (a t,n ) of the form () is to find a linear reccurence, and convert it into a differential equation for the generating series G t (z). The singularities of G t (z) 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 (a t,n ) follows from Wilf-Zeilberger s constructive theorem, and its computer implementation; see [Z, WZ, PWZ, PR, 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 G t (z), and then discard all but a small part of it in order to determine the singularities Σ t of G t (z). The main result of the paper is a construction of a finite set S t of algebraic numbers directly from the summand t n,k,...,k r, which we conjecture that it includes the set Σ t. We give a proof of our conjecture in some special cases, as well as supporting examples. Our definition of the set Σ t is reminiscent of the Bethe ansatz, and is related to critical values of potential functions and additive K-theory. Before we formulate our conjecture let us give an instructive example. Example.3. Consider the Apery sequence (a n ) defined by n ( ) 2 ( ) 2 n n + k (3) a n =. k k k=0 It turns out that (a n ) satisfies a linear recursion relation with coefficients in Q[n] (see [WZ2, p.74]) s=0 c λ,s n s (4) (n + 2) 3 a n+2 (2n + 3)(7n 2 + 5n + 39)a n+ + (n + ) 3 a n = 0 for all n N, with initial conditions a 0 =, a = 5. It follows that G(z) is holonomic; i.e., it satisfies a linear ODE with coefficients in Q[z]: (5) z 2 (z 2 34z + )G (z) + 3z(2z 2 5z + )G (z) + (7z 2 2z + )G (z) + (z 5)G(z) = 0 with initial conditions G(0) =, G (0) = 5, G (0) = 46. This implies that the possible singularities of G(z) are the roots of the equation: (6) z 2 (z 2 34z + ) = 0. Thus, G(z) has analytic continuation as a multivalued function in C \ {0, , 7 2 2}. Since the Taylor series coefficients of G(z) at z = 0 are positive integers, and G(z) is analytic at z = 0, it follows that G(z) has a singularity inside the punctured unit disk. Thus, G(z) is singular at By Galois invariance, it is also singular at The proof of Corollary.2 implies that (a n ) has an asymptotic expansion of the form: (7) a n ( ) n n 3/2 s=0 c s n s + (7 2 2) n n 3/2 s=0 c 2s n s

4 4 STAVROS GAROUFALIDIS for some constants c s, c 2s C with c 0 c A final calculation shows that c 0 = π 3/ , c 20 = π 3/ Our paper gives an ansatz that quickly produces the numbers 7 ± 2 2 given the expression (3) of (a n ), 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 c s0 for s = 0, in terms of the expression (3)..4. Hypergeometric terms. We have already mentioned multisums of balanced hypergeometric terms. Let us define what those are. Definition.4. An r-dimensional balanced hypergeometric term t n,k (in short, balanced term, also denoted by t) in variables (n, k), where n N and k = (k,...,k r ) Z r, is an expression of the form: r J (8) t n,k = C0 n A j (n, k)! ǫj i= C ri i where C i are algebraic numbers for i = 0,...,r, ǫ j = ± for j =,...,J, and A j are integral linear forms in (n, k) that satisfy the balance condition: (9) ǫ j A j = 0. Remark.5. An alternative way of encoding a balanced term is to record the vector (C 0,..., C r ), and the J (r + 2) matrix of the coefficients of the linear forms A j (n, k), and the signs ǫ j for j =,...,r..5. Balanced terms, generating series and singularities. Given a balanced term t, we will assign a sequence (a t,n ) and a corresponding generating series G t (z) to a balanced term t, and will study the set Σ t of singularities of the analytic continuation of G t (z) in the complex plane. Let us introduce some useful notation. Given a linear form A(n, k) in variables (n, k) where k = (k,...,k r ), and i = 0,...,r, let us define (0) v i (A) = a i, v 0 (A) = a 0, where A(n, k) = a 0 n + For w = (w,..., w r ) we define: () A(w) = a 0 + r a i w i. i= r a i k i. Definition.6. Given a balanced term t as in (8), define its Newton polytope P t by: (2) P t = {w R r A j (w) 0 forj =,..., J} R r. We will assume that P t is a compact rational convex polytope in R r with non-empty interior. It follows that for every n N we have: (3) support(t n,k ) = np t Z r. Definition.7. Given a balanced term t consider the sequence: (4) a t,n = k np t Z r t n,k (the sum is finite for every n N) and the corresponding generating function: (5) G t (z) = a t,n z n Q[[z]]. n=0 i=

5 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 5 Here Q denote the field of algebraic numbers. Let Σ t denote the finite set of singularities of G t, and E t = Q(Σ t ) denote the corresponding number field, following Corollary.2. Remark.8. Notice that t determines G t 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.9. The balance condition of Equation (9) is imposed so that for every balanced term t the corresponding sequence (a t,n ) grows at most exponentially. This follows from Stirling s formula (see Corollary 2.) and it implies that the power series G t (z) is the germ of an analytic function at z = 0. Given a proper hypergeometric term t n,k in the sense of [WZ], we can find α Q and ǫ = ± so that t n,k (αn)! ǫ is a balanced term..6. The definition of S t and K t. Let us observe that if t is a balanced term and is a face of its Newton polytope P t, then t is also a balanced term. Definition.0. Given a balanced term t as in Equation (8) consider the following system of Variational Equations: J (6) C i A j (w) ǫjvi(aj) = for i =,...,r in the variables w = (w,..., w r ). Let X t denote the set of complex solutions of (6), with the convention that when r = 0 we set X t = {0}, and define (7) CV t = {C0 A j (w) ǫjv0(aj) w X t } (8) (9) j:a j(w) 0 S t = {0} face of Pt CV t K t = Q(S t ). Remark.. There are two different incarnations of the set CV t : it coincides with (a) the set of critical values of a potential function; see Theorem 5. (b) the evaluation of elements of an additive K-theory group under the entropy regulator function; see Theorem 6. It is unknown to the author whether X t is always a finite set. Nevertheless, S t is always a finite subset of Q; see Theorem 5. Equations (6) are reminiscent of the Bethe ansatz..7. The conjecture. Section.5 constructs a map: Balanced terms t (E t, Σ t ) via generating series G t (z) and their singularities, where E t is a number field and Σ t is a finite subset of E t. Section.6 constructs a map: Balanced terms t (K t, S t ) via solutions of polynomial equations. We are now ready to formulate our main conjecture. Conjecture. For every balanced term t we have: Σ t S t and consequently, E t K t..8. Partial results. Conjecture is known to hold in the following cases: (a) For 0-dimensional balanced terms: see Theorem 2. (b) For positive special terms: see Theorem 3. (c) For -dimensional balanced terms, see [Ga3]. Since the finite sets S t and Σ t that appear in Conjecture are in principle computable (as explained in Section.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 6j-symbols, and of interest to atomic physics and low dimensional topology.

6 6 STAVROS GAROUFALIDIS 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 t n,k is 0-dimensional balanced term as in (8), with k =. Then G t (z) is a hypergeometric series and Conjecture holds. When t is positive dimensional, the generating series G t (z) is no longer hypergeometric in general. To state our next result, recall that a finite sybset S Q of algebraic numbers is irreducible over Q if the Galois group Gal( Q, Q) acrs transitively on S. Definition.2. A special hypergeometric term t n,k (in short, special term) is an expression of the form: (20) t n,k = C n 0 r i= C ri i J ( ) Bj (n, k). C j (n, k) where C Q, L and B j and C j are integral linear forms in (n, k). We will assume that for every n N, the support of t n,k = 0 as a function of k Z r is finite. We will call such a term positive if C i > 0 for i = 0,...,r. Lemma.3. (a) A balanced term is the ratio of two special terms. In other words, it can always be written in the form: s ( ) ǫj (2) t n,k = C L(n,k) Bj (n, k) C j (n, k) for some integral linear forms B j, C j and signs ǫ j. (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: t n,k = C n 0 r i= C ri i J j:ǫ A j(n, k)! A(n, k)! A(n, k)! J j:ǫ j= A j(n, k)! where A(n, k) = A j (n, k) = A j (n, k). j:ǫ j:ǫ j= To illustrate part (a) of the above lemma for 0-dimensional balanced terms, we have: (30n)!n! (6n)!(0n)!(5n)! = ( 30n 6n )( )( ) 4n 5n. 0n 4n 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 t n,k such that Σ t \ {0} is irreducible over Q. Then, Σ t CV t S t and Conjecture holds. In a forthcoming publication we will give a proof of Conjecture for -dimensional balanced terms; see [Ga3]. Let us end this section with an inverse type (or geometric realization) problem. Problem.4. Given λ Q, does there exist a special term t so that λ S t?

7 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 7.9. Laurent polynomials: a source of special terms. This section, which is of independent interest, associates an special term t F to a Laurent polynomial F with the property that the generating series G tf (z) is identified with the trace of the resolvant R F (z) of F. Combined with Theorem, this implies that R F (z) is a G-function. If F M N ( Q[x ±,..., x± d ]) is a square matrix of size N with entries Laurent polynomials in d variables, let Tr(F) denote the constant term of its usual trace. The moment generating series of F is the power series (22) G F (z) = Theorem 4. (a) For every F Tr(F n )z n. n=0 Q[x ±,...,x± d ] there exists a special term t F so that (23) G F (z) = G tf (z). Consequently, G F (z) is a G-function. (b) The Newton polytope P tf depends is a combinatorial simplex which depends only on the monomials that appear in F and not on their coefficients. (c) For every F M N ( Q[x ±,...,x± d ]), G F(z) is a G-function. We thank C. Sabbah for providing an independent proof of part (c) when N = using the regularity of the Gauss-Manin connection. Compare also with [DvK]..0. Plan of the proof. In Section 2, we introduce a potential function associated to a balanced term t and we show that the set of its critical values coincides with the set S t that features in Conjecture. This also implies that S t is finite. In Section 3 we assign elements of an extended additive K-theory group to a balanced term t, and we show that the set of their values (under the entropy regulator map) coincides with the set S t that features in Conjecture. 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. In Section 5 we give several examples that illustrate Conjecture. In Section 6 we study a special case of Conjecture, with input a Laurent polynomial in many commuting variables... 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 The author wishes to thank D. Zeilberger for his hospitality. 2. Balanced terms and potential functions 2.. 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 n! (see [O]): (24) log n! n log n n + 2 log n + 2 log(2π) + O ( n For x > 0, we can define x! = Γ(x + ). The Stirling formula implies that for a > 0 fixed and n N large, we have: ( ) log n (25) log(an)! = (n log n n)a + a log(a) + O n The next corollary motivates our definition of the potential function. ),

8 8 STAVROS GAROUFALIDIS Corollary 2.. For every balanced term t as in (8) and every w in the interior of P t we have: (26) t n,nw = e nvt(w)+o(log n n ). where the potential function V t is defined below. Definition 2.2. Given a balanced term t as in (8) define its corresponding potential function V t by: (27) V t (w) = C(w) + where w = (w,..., w r ), ǫ j A j (w)log(a j (w)) (28) C(n, k) = log C 0 n + log C k +... log C r k r, and (29) C(w) = log C 0 + log C w +... log C r w r. V t is a multivalued analytic function on the complement of the linear hyperplane arrangement C r \ A t, where (30) A t = {w C r In fact, let J A j (w) = 0}. (3) : Ĉr C r \ A t denote the universal abelian cover of C r \ A t. The next theorem relates the critical points and critical values of V t with the sets X t and CV t from Definition.0. Theorem 5. (a) For every balanced term t, (X t ) coincides with the set of critical points of V t. (b) If w is a critical point of V t, then J (32) e Vt(w) = C v0(l) A j (w) ǫjv0(aj) = C v0(l) j:a j(w) 0 A j (w) ǫjv0(aj). Thus CV t coincides with the exponential of the set of the negatives of the critical values of V t. (c) For every face of the Newton polytope of t we have: V (t ) = (V t ). (d) S t is a finite subset of Q and K t is a number field Proof of Theorem 5. Let us fix a balanced term t as in (8) and a face of its Newton polytope P t. Without loss of generality, assume that = P t. Since (33) d (xlog(x)) = log(x) + dx it follows that for every i =,...,r and every j =,...,J we have: (34) w i A j (w)log(a j (w)) = v i (A j )log(a j (w)) + v i (A j ). Adding up with respect to j, using the balancing condition of Equation (9), and the notation of Section.5 applied to the linear form of Equation (28), it follows that

9 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 9 w i V t (w) = log C i + = log C i + ǫ j v i (A j )log(a j (w)) + ǫ j v i (A j )log(a j (w)). ǫ j v i (A j ) This proves that the critical points w = (w,..., w r ) of V t are the solutions to the following system of Logarithmic Variational Equations: (35) log C i + ǫ j v i (A j )log(a j (w)) Z() for i =,...,r where, for a subgroup K of (C, +) and an integer n Z, we define (36) K(n) = (2πi) n K. Exponentiating, it follows that w satisfies the Variational Equations (6), and concludes the proof of part (a). For part (b), we will show that if w is a critical point of V t, the corresponding critical value is given by: (37) V t (w) = log C 0 + ǫ j v 0 (A j )log(a j (w)) C/Z(). Exponentiating, we deduce the first equality of Equation (32). The second equality follows from the fact that A j (w) 0 for all critical points w of V t. To show (37), observe that for any linear form A(n, k) we have: r A(w) = v 0 (A)w 0 + v i (A)w i. Suppose that w satisfies the Logarithmic Variational Equations (35). Using the definition of the potential function, and collecting terms with respect to w,...,w r it follows that V t (w) = log C 0 + = log C 0 + ǫ j v 0 (A j )log(a j (w)) + ǫ j v 0 (A j )log(a j (w)). i= r i= w i log C i + ǫ j v i (A j )log(a j (w)) This concludes part (b). Part (c) follows from set-theoretic considerations, and part (d) follows from the following facts: (i) an analytic function is constant on each component of its set of critical points, (ii) the set of critical points are the complex points of an affine variety defined over Q by (6), (iii) every affine variety has finitely many connected components. This concludes the proof of Theorem Balanced terms, the entropy function and additive K-theory In this section we will assign elements of an extended additive K-theory group to a balanced term, and using them, we will identify our finite set CV t from Definition (.0) with the values of the constructed elements under a regulator map; see Theorem 6.

10 0 STAVROS GAROUFALIDIS 3.. A brief review of the entropy function and additive K-theory. In this section we will give a brief summary of an extended version of additive K-theory and the entropy function following [Ga2], and motivated by [Ga]. This section is independent of the rest of the paper, and may be skipped at first reading. Definition 3.. Consider the entropy function Φ, defined by: (38) Φ(x) = xlog(x) ( x)log( x). for x (0, ). Φ(x) is a multivalued analytic function on C \ {0, }, given by the double integral of a rational function as follows from: (39) Φ (x) = x x. For a detailed description of the analytic continuation of Φ, we refer the reader to [Ga2]. Let Ĉ denote the universal abelian cover of C. In [Ga2, Sec..3] we show that Φ has an analytic continuation: (40) Φ : Ĉ C. In [Ga2, Def..7] we show that Φ satisfies three 4-term relations, one of which is the analytic continuation of (48). The other two are dictated by the variation of Φ along the cuts (, ) and (, 0). Using the three 4-term relations of Φ, we introduce an extended version β 2 (C) in [Ga2, Def..7]: Definition 3.2. The extended group β 2 (C) is the C-vector space generated by the symbols x with x = (z; p, q) Ĉ, subject to the extended 4-term relation: (4) x 0 ; p 0, q 0 x ; p, q + ( x 0 ) ; p 2, q 2 ( x ) ; p 3, q 3 = 0 x 0 x for ((x 0 ; p 0, q 0 ),..., (x 3 ; p 3, q 3 )) 4T, and the relations: x x 0 (42) (43) x; p, q x; p, q = x; p, q 2 x; p, q 2 x; p, q x; p, q = x; p 2, q x; p 2, q for x C, p, q, p, q 2Z. Since the three 4-term relations in the definition of β 2 (C) are satisfied by the entropy function, it follows that Φ gives rise to a regulator map: (44) R : β 2 (C) C. For a motivation of the extended group β 2 (C) and its relation to additive (i.e., infinitesimal) K-theory and infinitesimal polylogarithms, see [Ga2, Sec..] and references therein Balanced terms and additive K-theory. In this section, it will be more convenient to use the presentation (2) of balanced terms. In this case, we have: (45) t n,k = C n 0 r i= C ki i J B j (n, k)! ǫj C j (n, k)! ǫj (B j C j )(n, k)! ǫj. The Stirling formula motivates the constructions in this section. Indeed, we have the following:

11 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS Lemma 3.3. For a > b > 0, we have: ( ) b (46) aφ = a log(a) b log(b) (a b)log(a b). a and (47) ( ) an e naφ( ba ) bn a 2b(a b)πn ( + O ( )). n 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 a, b, a + b (0, ), Φ satisfies the 4-term relation: ( ) ( ) a b (48) Φ(b) Φ(a) + ( b)φ ( a)φ = 0. b a Proof. The 4-term relation follows from the associativity of the multinomial coefficients (49) ( )( ) ( )( ) α + β + γ β + γ α + β + γ α + γ = = α β β α (α + β + γ)! α! β! γ! applied to (α, β, γ) = (an, bn, cn) and Lemma 3.3, and the specialization to a+b+c =. In fact, the 4-term relation (48) and a local integrability assumption uniquely determines Φ up to multiplication by a complex number. See for example, [Da] and [AD, Sec.5.4,p.66]. Corollary 3.5. If t is a balanced term as in (2), then its potential function is given by: ( ) Cj (w) (50) V t (w) = C(w) + ǫ j B j (w)φ. B j (w) Consider the complement C r \ A t of the linear hyperplane arrangement given by: (5) A t = {w Cr Let J B j (w)c j (w)(b j (w) )(B j (w) C j (w)) = 0}. (52) : Ĉr C r \ A t denote the universal abelian cover of C r \ A t. For j =,...,r, the functions B j(w), C j (w)/b j (w) have analytic continuation: C j B j : Ĉr C, : B Ĉr Ĉ j It follows that the potential function given by (50) has an analytic continuation: (53) V t : Ĉr C. Let C t denote the set of critical points of V t. Definition 3.6. For a balanced term t as in (2), we define the map: (54) β t : C t β 2 (C) by: β t (w) = ǫ j B j (w) C j(w) B j (w)

12 2 STAVROS GAROUFALIDIS Theorem 6. For every balanced term t, we have a commutative diagram: C t β t β2 (C) X t C 0e V t e R CV t 3.3. Proof of Theorem 6. We begin by observing that the analytic continuation of the logarithm function gives an analytic function: Log : Ĉ C. The proof of part (a) of Theorem 5 implies that w C t if and only if w satisfies the Logarithmic Variational Equations: (55) log C i + ǫ j (v i (B j )Log(B j (w)) v i (C j )Log(C j (w)) v i (B j C j )Log((B j C j )(w))) = 0 for i =,...,r. Exponentiating, this implies that C t = (X t ). The proof of part (b) of Theorem 5 implies that if w C t, then: V t (w) = log C 0 + ǫ j (v 0 (B j )Log(B j (w)) v 0 (C j )Log(C j (w)) v 0 (B j C j )Log((B j C j )(w))). On the other hand, if w C t, we have: R(β t (w)) = = ( ) Cj (w) ǫ j B j (w)φ B j (w) ǫ j (B j (w)log(b j (w)) C j (w)log(c j (w)) (B j (w) C j (w))log((b j C j )(w))) where the last equality follows from the analytic continuation of (46). Expanding the linear forms B j, C j, and B j C j with respect to the variables w i for i = 0,...,r, and using the Logarithmic Variational Equations (55), (as in the proof of part (b) of Theorem 5), it follows that Thus, R(β t (w)) = ǫ j (v 0 (B j )Log(B j (w)) v 0 (C j )Log(C j (w)) v 0 (B j C j )Log((B j C j )(w))). This concludes the proof of Theorem 6. V t (w) = log C 0 R(β t (w)). 4. Proof of Theorems 2 and Proof of Theorem 2. A 0-dimensional balanced term is of the form: t n = C n 0 J (b j n)! ǫj

13 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 3 where b j N, ǫ j = ± for j =,...,J satisfying J ǫ jb j = 0. Since Z 0 = {0}, it follows that a t,n = t n is so-called closed form. The Newton polytope of t is given by P t = {0} R 0. In addition, A j (w) = b j and v 0 (A j ) = b j for j =,...,J. By definition, we have X t = {0}, and (56) CV t = {C 0 J b ǫjbj j }. On the other hand, G t (z) is a hypergeometric series with singularities {0, CV t }; see for example [O, Sec.5] and [No]. The result follows 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 P t. See also [Kn, Sec.5..4]. Suppose that t n,k 0 for all n, k. Recall the corresponding polytope P t R r and consider the restriction of the potential function to P t : V t : P t R. It is easy to show that Φ(x) > 0 for x (0, ). This is illustrated by the plot of the entropy function for x [0, ]: It follows that the restriction of the potential function on P t is nonnegative and continuous. By compactness it follows that the function achieves a maximum in ŵ in the interior of P t. It follows that for every k np t Z r we have: 0 t n,k t n,nŵ Summing up over the lattice points k np t Z r, and using the fact that the number of lattice points in a rational convex polyhedron (dilated by n) is a polynomial function of n, it follows that there exist a polynomial p(n) Q[n] so that for all n N we have: t n,nŵ a t,n p(n)t n,nŵ Using Corollary 2., it follows that there exist polynomials p (n), p 2 (n) Q[n] so that for all n N we have: p (n)e nvt(ŵ) a t,n p 2 (n)e nvt(ŵ) This implies that G t (z) has a singularity at z = e Vt(ŵ) > 0. Since the maximum lies in the interior of P t, it follows that ŵ is a critical point of V t. Thus, ŵ S t and consequently, e Vt(ŵ) CV t. If in addition Σ t \ {0} is irreducible over Q, it follows that the Galois group Gal( Q/Q) acts transitively on Σ t. This implies that Σ t CV t. Remark 4.. When t n,k 0 for all n, k, then G t (z) has a singularity at ρ > 0 where /ρ is the radius of convergence of G t (z). This is known as Pringsheim s theorem, see [Ti, Sec.7.2].

14 4 STAVROS GAROUFALIDIS 5. Some examples 5.. A closed form example. As a warm-up example, consider the -dimensional special term ( ) n n! t n,k = = k k!(n k)! whose corresponding sequence is closed form and the generating series is a rational function: a t,n = 2 n G t (z) = 2z In that case Σ t = {/2} and E t = Q. To compare with our ansatz, the Newton polytope is given by: The Variational Equations (6) are: in the variable w = w, with solution set P t = [0, ] R. w( w) = X t = {/2} Thus, CV t = { w = /2} = {/2}. ( w) For the other two faces 0 = {0} and = {} of the Newton polytope P t, the restriction is a 0-dimensional balanced term. Equation (56) gives that Thus, This implies that t 0 = t n,k k=0 =, t = t n,k k=n =. CV t 0 = CV t = {}. S t = {0,, /2} K t = Q, confirming Conjecture. For completeness, the potential function is given by: V t (w) = Φ(w) The Apery sequence. As an illustration of Conjecture and Theorem 3, let us consider the special term ( ) 2 ( ) 2 ( ) 2 n n + k (n + k)! (57) t n,k = = k k k! 2 (n k)! and the corresponding sequence (3). Equation (5) implies that the singularities of G(z) are a subset of the roots of the equation z 2 (z 2 34z + ) = 0. In addition, a nonzero singularity exists. Thus, we obtain that (58) {0, , 7 2 2} Σ t {0, , 7 2 2}, E t = Q( 2). Thus E t is a quadratic number field of type [2, 0] and discriminant 8. On the other hand, the Newton polytope is given by: (59) P t = [0, ] R.

15 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 5 The Variational Equations (6) are: (60) in the variable w = w, with solution set ( w w ) 2 + w = w X t = {/ 2, / 2}. Thus, ( ) 2 w CV t = { w = ±/ 2} = { , 7 2 2} + w For the other two faces 0 = {0} and = {} of the Newton polytope P t the restriction is a 0-dimensional balanced term: ( ) 2 (2n)! t 0 = t n,k k=0 =, t = t n,k k=n = n! 2 Equation (56) implies that CV t 0 = {}, CV t = {6}. Therefore, S t = {0,, 6, , 7 2 2} K t = Q( 2), confirming Conjecture. For completeness, the potential function is given by: ( ) w V t (w) = 2Φ(w) + 2( + w)φ. + w 5.3. An example with critical points at the boundary. In this example we find critical points at the boundary of the Newton polytope even though the balanced term is positive. This shows that Theorem 3 is sharp. Consider the balanced term (6) t n,k = ( (n k)!k! n ) = n! k and the corresponding sequence (a t,n ). The zb.m implementation of the WZ algorithm (see [PR, PR2]) gives that (a t,n ) satisfies the inhomogeneous recursion relation: (62) 2(n + )a n+ + (n + 2)a n = 2 2n for all n N with initial conditions a 0 =. It follows that (a n ) satisfies the homogeneous recursion relation: (63) 2(n + 3)a n+3 + (2 + 5n)a n+2 4(n + 2)a n+ + (n + 2)a n = 0 for all n N with initial conditions a 0 =, a = 2, a 2 = 5/2. The corresponding generating series G t (z) satisfies the inhomogeneous ODE: (64) (z ) 2 (z 2)f (z) + 2(z ) 2 f(z) + 2 = 0 with initial conditions f(0) =. We can convert (64) into a homogeneous ODE by differentiating once. It follows that (65) Σ t {, 2}, E t = Q. On the other hand, P t = [0, ] R. The Variational Equations (6) are: w w =

16 6 STAVROS GAROUFALIDIS in the variable w = w with solution set X t = {/2} and CV t = { w w = 2 } = {2}. For the other two faces 0 = {0} and = {} of the Newton polytope P t we have: Thus, t 0 = t n,k k=0 =, t = t n,k k=n =. CV t 0 = CV t = {}. This implies that S t = {0,, 2} K t = Q, confirming Conjecture. For completeness, the potential function is given by: V t (w) = Φ(w). 6. Laurent polynomials and special terms 6.. Proof of Theorem 4. In this section we will prove Theorem 4. Consider F decompose F into a sum of monomials with coefficients ± Q[x,..., x± ]. Let us d (66) F = α A where A is the finite set of monomials of F, and where for every α = (α,..., α d ) we denote x α = d Let (67) r = A c α x α denote the number of monomials of F. Recall that n+ n = {(y,..., y n+ ) R n+ y j =, y i 0, i =,...,n + } P α A kα=n xαj j. denotes the standard n-dimensional simplex in R n+. Part (a) of Theorem 4 follows easily from the multinomial coefficient theorem. Indeed, for every n N we have: F n n! = α A k c kα α α! xkαα It follows that for every n N we have: = P α A kα=n n! α A k α! α A α A c kα α xp α A kαα where Tr(F n ) = = P α A kα=n,p α A kαα=0 k np tf Z r t F,n,k n! α A k α! α A c kα α (68) t F,n,k = and the Newton polygon P tf is given by n! α A k α! α A c kα α

17 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 7 (69) P tf = r W where (70) W = {(x α ) R r α Ax α α = 0} is a linear subspace of R r. To prove part (b) of Theorem 4, let us assume that the origin is in the interior of the Newton polytope of F. Such F are also called convenient in singularity theory; [Kou]. If F is not convenient, we can replace it with its the restriction F f to a face f of its Newton polytope that contains the origin and observe that Tr((F f ) n ) = Tr(F n ). When F is convenient, it follows that W has dimension r d and W C o, where C o is the interior of the cone C which is spanned by the coordinate vectors in R r. (b) follows from Lemma 6.3 below. For part (c) of Theorem 4, fix F M N ( Q[x ±,..., x± d ]), G F (z). Recall that for an invertible matrix A we have A = det(a) Cof(A), where Cof(A) is the co-factor matrix. It follows that (7) F n z n = (I zf) = n=0 where Ad(I zf) M N ( Q[x ±,..., x±, z]). Now d det(i zf) = α S where S is a finite set, c α Q and d α N \ {0}. Thus, det(i zf) = n=0 P α S kα=n Ad(I zf) det(i zf) c α z dα x α n! α S k α! zp α S dαkα α S x P α S kαα Substituting the above into Equation (7) and taking the constant term, it follows that there exists a finite set S and a finite collection t (j) F of balanced terms for j S such that for every n N we have: Tr(F n ) = j S a t (j) F,n It follows that the moment generating series G F (z) (defined in Equation (22)) is given by: G F (z) = G (j) t (z). F j S Since G (j) t (z) is a G-function (by Theorem ), and the set of G-functions is closed under addition (see [An]), F this concludes the proof Theorem 4. Remark 6.. The balanced terms t (j) F in the above proof use affine linear forms, rather than linear ones. In other words, they are given by (72) t n,k = C n 0 r i= C ri i J A j (n, k)! ǫj where C i are algebraic numbers for i = 0,...,r, ǫ j = ± for j =,...,J, and A j are affine linear forms, given by A j (n, k) = A lin j (n, k) + b j where A lin j (n, k) are linear forms that satisfy the balance condition (73) ǫ j A lin j = 0.

18 8 STAVROS GAROUFALIDIS Theorem remains true for such balanced terms. For an balanced term t of the form (72) consider the balanced term t lin defined by: (74) t lin n,k = C n 0 We define S t = S tlin. r i= C ri i J A lin j (n, k)! ǫj Remark 6.2. In the notation of Theorem 4, P tf is a simple polytope and the associated toric variety is projective and has quotient singularities; see [Fu]. In other words, the stabilizers of the torus action on the toric variety are finite abelian groups. The following lemma was communicated to us by J. Pommersheim. Lemma 6.3. If W is a linear subspace of R n of dimension s, and W C o, where C is the cone spanned by the n coordinate vectors in R n, then n W is a combinatorial simplex in V. Proof. We can prove the claim by downward induction on s. When s = n, W is a hyperplane. Write D n = C H where H is the hyperplane given by n i= x i = 0. Observe that C is a simplicial cone and the intersection C W is a simplicial cone in W. Since the intersection of a simplicial cone inside C with H is a simplicial cone, it follows that D n W = (C W) H is a simplicial cone. This proves the claim when s = n. A downward induction on s concludes the proof The Newton polytope of a Laurent polynomial and its associated balanced term. In a later publication we will study the close relationship between the Newton polytope of a Laurent polynomial F and the Newton polytope of the corresponding special term t F. In what follows, fix a Laurant polynomial F as in (66) and its associated balanced term t F as in (68). With the help of the commutative diagram of Proposition 6.5 below, we will compare the extended critical values of F with the set S tf. Keep in mind that [DvK] use the Newton polytope of F, whereas our ansatz uses the Newton polytope of t F. Consider the polynomial map ± ± (75) φ : Q[w α α A] Q[x,..., x± d ] given by φ(w α ) = x α. Its kernel ker(φ) is a monomial ideal. Adjoin the monomial w α0 to A (if it not already there), where α 0 = 0 Z d, and consider the homogenous ideal ker h (φ), where the degree of w α is α. Lemma 6.4. The Variational Equations (6) for the balanced term t F are equivalent to the following system of equations: α A x pα α c pα α = for (wα pα) (φ) (76) α A x αα = 0 α A x α =. in the variables (x α ) α A. Proof. This follows easily from the description of the Newton polytope P tf and the shape of the balanced term t. Consider the rational function (77) Ψ F : (C ) d \ F (0) (C ) r, ( cα u α ) u Ψ F (u) = F(u) Observe that the image of Ψ F lies in the complex affine simplex {(x α ) (C ) r α Ax α = }. α A

19 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 9 Proposition 6.5. (a) If u = (u,..., u d ) is a critical point of the restriction of F on (C ) d with nonvanishing critical value, then Ψ F (u) satisfies the Variational Equations (6) for the maximal face P tf of the Newton polytope P tf of t P. (b) Restricting to those critical points, we have a commutative diagram: Critical points of F on (C ) d Ψ F X tf,p tf (78) F Map (7) Critical values of F S tf,p tf (c) The top horizontal map of the above diagram is - and onto. Proof. For (a) we will use the alternative system of Variational Equations given in (76). Suppose that u = (u,...,u d ) is a critical point of F with nonzero critical value, and let (w α ) α A denote the tuple (c α u α /F(u)) α A. We need to show that (w α ) α A is a solution to Equations (76). Consider a point (p α ) α A P tf. Then, we can verify the first line of Equation (76) as follows: α A wα pα c pα α = α A ( cα u α F(u) ) pα c pα α = u P α A pαα F(u) P α A pα = since (w pα α ) α A in in the kernel of φ. To verify the second line of Equation (76), recall that z z (z k ) = kz k. Since u is a critical point of F, it follows that for every i =,...,d we have: u i ui F(u) = α Av i (α)c α u α = 0 where v i (α) is the ith coordinate of α. It follows that α A w αα = 0, which verifies the second line of (76). To verify the third line, we compute: c α u α F(u) = c α u α = F(u) F(u) F(u) =. α A α A (b) and (c) follow from similar computations. Let us end this section with an example that illustrates Lemma 6.4 and Proposition 6.5. Example 6.6. Consider the Laurent polynomial Its Newton polytope is given by F(x, x 2 ) = ax + bx + cx 2 + dx 2 + fx x 2 + g

20 20 STAVROS GAROUFALIDIS With the notation from the previous section, we have d = 2, r = 6. Moreover, for every n N, we have It follows that F n = k + +k 6=n where the Newton polytope P tf R 6 is given by n! k!... k 6! ak b k2 c k3 d k4 f k5 g k6 x k k2+k5 x k3 k4+k5 2. Tr(F n ) = n! b k!... k 6! ak k2 c k3 d k4 f k5 g k6 (k,...,k 6) np tf Z 6 (79) P tf = {(x,..., x 6 ) R 6 x x 2 + x 5 = 0, x 3 x 4 + x 5 = 0, x + + x 6 =, x i 0} We can use the variables (x, x 3, x 5 ) to parametrize P tf as follows: (80) P tf = {(x, x 3, x 5 ) R 3 2x + 2x 3 + 3x 5, x i 0} This confirms that P tf is a combinatorial 2-dimensional simplex. If (k,..., k 6 ) P tf, then (k 2, k 4, k 6 ) = (k + k 5, k 3 + k 5, n 2k 2k 3 3k 5 ), and Tr(F n ) = 2k +2k 3+2k 5=n The Variational Equations (6) are: (8) n! k!(k + k 5 )!k 3!(k 3 + k 5 )!k 5!(n 2k 2k 3 3k 5 )! ( ab g 2 ab w (w + w 5 )( 2w 2w 3 3w 5 ) 2 g 2 = cd w 3 (w 3 + w 5 )( 2w 2w 3 3w 5 ) 2 g 2 = bdf (w + w 5 )(w 3 + w 5 )w 5 ( 2w 2w 3 2w 5 ) 3 g 3 = in the variables (w, w 3, w 5 ). Reintroducing the variables w 2, w 4 and w 6 defined by the Variational Equations become ) k ( cd g 2 w 2 = w + w 5, w 4 = w 3 + w 5, w 6 = 2w 2w 3 2w 5 ) k3 ( ) k5 bdf g 3 g n. (82) w 2 = w + w 5 w 4 = w 3 + w 5 (83) w w 2 w 2 6 w 3 w 4 w 2 6 w 6 = 2w 2w 3 2w 5 ab g 2 = cd g 2 = bdf w 2 w 4 w 5 w6 3 g 3 = On the other hand, w α0 = w 6 and ker(φ) is generated by the relations w 6 =, w w 2 =, w 3 w 4 = and w 2 w 4 w 5 = which lead to the homogenous relations w w 2 w 2 6 =, w 3 w 4 w 2 6 = and w 2 w 4 w 5 w 3 6 for ker h (φ) that appear in the above Variational Equations. This illustrates Lemma 6.4. Suppose now that u = (u, u 2 ) is a critical point of F. Then, u satisfies the equations:

21 AN ANSATZ FOR THE ASYMPTOTICS OF HYPERGEOMETRIC MULTISUMS 2 (84) (85) F u = a b u 2 + fu 2 = 0 F u 2 = c d u 2 + fu = 0. 2 The map Ψ F is defined by Ψ F (u, u 2 ) = (w,...,w 6 ) where w = au F(u), w 2 = b u F(u), w 3 = cu 2 F(u), w 4 = d u 2 F(u), w 5 = fu u 2 F(u), w 5 = g F(u). If u = (u, u 2 ) satisfies Equations (84), (85), it is easy to see that w = Ψ F (u) satisfies the Variational Equations (82) and (83). For example, we have and This illustrates Proposition 6.5. w 3 + w 5 w 4 = cu 2 F(u) + fu u 2 F(u) d u 2 F(u) = u ( 2 c + fu d ) F(u) u 2 = 0 2 cd w 3 w 4 w6 2 g 2 = cu 2 d F(u) u 2F(u) References cd ( ) 2 g g 2 =. F(u) [An] Y. Andre, G-functions and geometry, Aspects of Mathematics, E3. Braunschweig, 989. [An2] Y. André, Séries Gevrey de type arithmétique. I. Théorèmes de pureté et de dualité, Ann. of Math. (2) 5 (2000) [AD] J. Aczél and J. Dhombres, Functional equations in several variables, Encyclopedia of Mathematics and its Applications 3 Cambridge University Press, Cambridge, 989. [BT] G. Birkhoff and W. Trjitzinsky, Analytic theory of singular difference equations, Acta Math. 60 (932) 89. [CC] D.V. Chudnovsky and G.V. Chudnovsky, Applications of Padé approximations to the Grothendieck conjecture on linear differential equations, in Number theory Lecture Notes in Math. 35 Springer-Verlag (985) [CG] O. Costin and S. Garoufalidis, Resurgence of the Kontsevich-Zagier power series, Annales de l Institut Fourier, in press. [Da] Z. Daróczy, Generalized information functions, Information and Control 6 (970) [DvK] J.J. Duistermaat and W. van der Kallen, Constant terms in powers of a Laurent polynomial, Indag. Math. 9 (998) [DGS] B. Dwork, G. Gerotto and F.J. Sullivan, An introduction to G-functions, Annals of Mathematics Studies 33 Princeton University Press, 994. [FS] P. Flajolet and R. Sedgewick, Analytic combinatorics: functional equations, rational and algebraic functions, electronic book. [Fu] W. Fulton, Introduction to toric varieties, Annals of Mathematics Studies 3 Princeton University Press 993. [Ga] S. Garoufalidis, q-terms, singularities and the extended Bloch group, preprint 2007, arxiv: [Ga2], An extended version of additive K-theory, Journal of K-theory, in press. [Ga3], Resurgence of -dimensional hypergeometric multisums, preprint [Ga4], G-functions and multisum versus holonomic sequences, preprint 2007 arxiv: [GV] and R. van der Veen, Asymptotics of 6j-symbols, preprint [Ju] R. Jungen, Sur les séries de Taylor n ayant que des singularités algébrico-logarithmiques sur leur cercle de convergence, Comment. Math. Helv. 3 (93) [Ka] N.M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin, IHES Publ. Math. No. 39 (970) [Kn] D. Knuth, The art of computer programming, vol 3, Sorting and searching. Addison-Wesley Publishing Co., 973. [KM] M. Kontsevich and Yu. Manin, Gromov-Witten classes, quantum cohomology, and enumerative geometry, in Mirror symmetry, II AMS/IP Stud. Adv. Math., (999) [KZ] and D. Zagier, Periods, in Mathematics unlimited 200 and beyond, (200) [Kou] A.G. Kouchnirenko, Polyèdres de Newton et nombres de Milnor, Invent. Math. 32 (976) 3. [O] F. Olver, Asymptotics and special functions, Reprint. AKP Classics. A K Peters, Ltd., Wellesley, MA, 997.

22 22 STAVROS GAROUFALIDIS [Ni] N. Nilsson, Some growth and ramification properties of certain integrals on algebraic manifolds, Ark. Mat. 5 (965) [No] N.E. Norlund, Hypergeometric functions Acta Math. 94 (955) [PR] P. Paule and A. Riese, A Mathematica q-analogue of Zeilberger s Algorithm Based on an Algebraically Motivated Approach to q-hypergeometric Telescoping, in Special Functions, q-series and Related Topics, Fields Inst. Commun., 4 (997) [PR2], Mathematica software: [PWZ] M. Petkovšek, H.S. Wilf and D.Zeilberger, A = B, A.K. Peters, Ltd., Wellesley, MA 996. [R-V] F. Rodriguez-Villegas, Integral ratios of factorials and algebraic hypergeometric functions, talk in Oberwolfach math.nt/ [Si] C.L. Siegel, Über einige Anwendungen diophantischer Approximationen, Abh. Preuss. Akad. Wiss. (929) 70. Reprinted in Gesammelte Abhandlungen, vol., no 6 (966) [St] R. Stanley, Enumerative combinatorics, Vol., 2. second edition, Cambridge University Press, Cambridge, 997. [Ti] E.C. Titchmarsh, The theory of functions, second edition, Oxford Univ. Press 939. [WZ] H. Wilf and D. Zeilberger, An algorithmic proof theory for hypergeometric (ordinary and q) multisum/integral identities, Inventiones Math. 08 (992) [WZ2] J. Wimp and D. Zeilberger, Resurrecting the asymptotics of linear recurrences, J. Math. Anal. Appl. (985) [Z] D. Zeilberger, A holonomic systems approach to special functions identities, J. Comput. Appl. Math. 32 (990) School of Mathematics, Georgia Institute of Technology, Atlanta, GA , USA, stavros address: stavros@math.gatech.edu