Log-concavity of asymptotic multigraded Hilbert series

We study the linear map sending the numerator of the rational function representing the Hilbert series of a module to that of its r-th Veronese submodule. We show that the asymptotic behaviour as r tends to infinity depends on the multidegree of the module and the underlying positively multigraded polynomial ring. More importantly, we give a polyhedral description for the asymptotic polynomial and prove that the coefficients are log-concave. In particular, we extend some results by Beck-Stapledon and Diaconis-Fulman beyond the standard graded situation.


INTRODUCTION
Although motivated by multigraded Hilbert series, our main result only involves linear operators on a multivariate power series. To be explicit, let A := [a 1 · · · a n ] be an integer (d × n)-matrix of rank d such that the only nonnegative vector in the kernel is the zero vector (i.e. a 1 , . . . , a n is an acyclic vector configuration; see [Zie,§6.2]). Equivalently, the rational function 1/ ∏ 1 j n (1 − t a j ) has a unique expansion as a multivariate formal power series; cf. Lemma 8.16 in [MS]. For each positive integer r, consider the linear operator Φ r induced by sifting out all terms with the exponent vector divisible by r in the power series expansion of a rational function. More precisely, Φ r acts on F(t) ∈ Z[t ±1 ] := Z[t 1 ,t −1 1 , . . .,t d ,t −1 d ] as follows: The goal of this article is to understand Φ r [F(t)] for r ≫ 0. To state our result, α : R n → R d denotes the linear map determined by A. The zonotope Z is the image under α of the unit hypercube [0, 1] n ⊂ R n . For each u ∈ Z d , set P(u) := α −1 (u) ∩ [0, 1] n . We say that the map α is degenerate if there exists u in the boundary of Z such that dim P(u) = n − d. By identifying the rational polytope P(u) with a translate P(u) + x lying in ker(A) = α −1 (0), the normalized volume vol n−d P(u) equals (n − d)! times the Euclidean volume of P(u) + x with respect to the lattice α −1 (0) ∩ Z n . Let m be the greatest common divisor of the maximal minors of A; in other words, the sublattice ZA ⊆ Z d generated by the columns of A has index m.
Moreover, the coefficients of K A (t) are log-concave, quasi-concave, and sum to m n−d (n − d)!.
When A = [1 · · · 1], Theorem 1.1 specializes to Theorem 5.1 in [DF2].  [BHVW,§4.3] or the mixed Eulerian numbers in [Pos,§16]. Nevertheless, the proof of Theorem 1.1 (see Step 1 in §2) implies that, for any has nonnegative coefficients that are both log-concave and quasi-concave. Since quasi-concavity is the multivariate version of unimodality (see Step 2 in §2), Theorem 1.1 generalizes those parts of Theorem 1.4 in [BW] and Theorem 1.2 in [BS] that do not make explicit reference to the roots of It is an open problem to effectively bound r 0 . Despite sharing most properties of the Eulerian polynomial, the Laurent polynomial K A (t) need not satisfy the multivariate analogues of real-zero univariate polynomials. For example, if then we have K A (t 1 ,t 2 ,t 3 ) = t 1 t 3 +t 2 which does not have the half-plane property; see Theorem 3.2 in [Bra]. Notably, the polynomial K A (t 1 ,t 2 ,t 3 ) = t 1 t 3 + t 2 is neither real stable nor Hurwitz stable. In contrast with [BW], [BS] or [DF2], we provide a new proof that the Eulerian numbers n i are log-concave and unimodal. We use polyhedral geometry and the Prékopa-Leindler inequality (a.k.a. the functional form of the Brunn-Minkowski inequality) to establish the log-concavity for the coefficients of K A (t); see Proposition 2.1. In view of [Sta2] and [Bre], our approach to log-concavity appears to be novel within algebraic combinatorics.
Our primary motivation for Theorem 1.1 comes from commutative algebra and algebraic geometry. The matrix A defines a positive Z d -grading on the polynomial ring S := C[x 1 , . . ., x n ] by setting deg(x j ) := a j for all 1 j n; see Definition 8.7 in [MS]. For any finitely generated Z d -graded Smodule M = w∈Z d M w , the Hilbert series of M can be expressed uniquely as a rational function of the form F(t)/ ∏ 1 j n (1 − t a j ) ; see Theorem 8.20 in [MS]. The numerator F(t) is the Poincaré polynomial or K-polynomial of M -it records the alternating sum of the multigraded Betti numbers for M; see Definition 8.21 and Theorem 8.23 in [MS]. Applying the operator Φ r to the Hilbert series of M yields the Hilbert series of the r-th Veronese submodule w∈Z d M rw . Theorem 1.1 shows that there exists a unique asymptotic K-polynomial for each M; see the first subsection in §3 for the important case in which F(1) = 0. Therefore, Theorem 1.1 gives some information about the solution to Problem 5.3 in [EL]. The results in both [EL] and this paper suggest that the asymptotic structure is surprisingly insensitive to the specific module.
Geometrically, the Z d -grading on S corresponds to the action of the torus (C * ) d on A n and a multigraded S-module M corresponds to a torus-equivariant sheaf on A n . In this setting, the numerator F(t) is the class in the equivariant K-theory of A n represented by the corresponding sheaf; see Theorem 8.34 in [MS]. Hence, Theorem 1.1 implies that the Veronese operator Φ r distinguishes the class K A (t) within equivariant K-theory. By taking quotients, we see that the toric variety A n //(C * ) d is equipped with a distinguished K-theory class K A (t). A geometric explanation for this distinguished K-theory class would be interesting.
Our secondary motivation comes from the theory of lattice point enumeration. Given lattice polytopes P 1 , . . . , P d in R n and nonnegative integers w 1 , . . . , w d , the number of lattice points in the Minkowski sum of the w i -dilates of the P i is denoted h-vector or δ -polynomial) for the collection P 1 , . . . , P d . Thus, Theorem 1.1 also yields a multivariate analogue of Corollary 1.3 in [BS].
Multivariate formal power series of the form F(t)/ ∏ 1 j n (1 − t a j ) also arise naturally in many other areas of mathematics. Perhaps the most ubiquitous source is the vector partition function ψ A : N d → N associated to A; ψ A (u) counts the number of nonnegative integer vectors x ∈ N n such that Ax = u. The generating series ∑ u ψ A (u) t u equals 1/ ∏ 1 j n (1 − t a j ) . As [Stu] indicates, vector partition functions appear in representation theory, approximation theory, and statistics. Reinterpreting Φ r in each of these areas will yield new insights into K A (t).
We end this section with two examples illustrating the necessity of the hypotheses in Theorem 1.1.
In particular, the map α is degenerate.

⋄
Further examples, open problems, and other connections are discussed in §3. The proof of Theorem 1.1 is given in §2.

Acknowledgements.
We thank Matthias Beck, Daniel Erman, and Alan Stapledon for stimulating our interest in asymptotic Hilbert series and providing feedback on a preliminary version of this document. We also thank Mike Roth for his valuable comments and insights. The computer software Macaulay2 [M2] was useful for generating examples. Both authors were partially supported by NSERC.

THE PROOF OF THE MAIN THEOREM
We divide the proof of Theorem 1.1 into three steps: Step 1 establishes that the limit exists and provides a polyhedral interpretation for K A (t), Step 2 uses the polyhedral interpretation to prove the log-concavity of the coefficients, and Step 3 gives a geometric explanation for the sum of the coefficients. For brevity, we write ∏ j := ∏ 1 j n and ∑ j := ∑ 1 j n . STEP 1.
To begin, we describe the matrix associated to the linear operator Φ r . Despite being defined via multivariate formal power series, Φ r may be understood in terms of a linear operator on Laurent polynomials. Specifically, let Ψ r ∈ End Z[t ±1 ] be the linear operator that discards the terms with exponent vectors that are not componentwise divisible by r and divides each of the remaining exponent vectors by r. Consider a Laurent polynomial . The linear operator Ψ r lifts to an endomorphism on multivariate formal power series. Applying Ψ r to this rational function yields which is a multivariate version of Lemma 3.2 in [BS]. To express Φ r as a matrix with respect to the monomial basis, set As v ∈ ZA, we also have C r (u, v) = 0 for all u ∈ ZA. However, the sublattice ZA ⊆ Z d has index m so mu ∈ ZA.

Hence, it is enough to consider
We next relate the integer coefficients C mr (u, v) to the normalized volume of the rational polytope P(u) := {x ∈ [0, 1] n : ∑ j x j a j = u} = α −1 (u) ∩[0, 1] n ⊆ R n . For any u ∈ Z d and any v ∈ ZA, there exists y and z ∈ Z n such that mu = α(y) and v = α(z). It follows that C mr (u, v) = (mr − 1)P(u) + 1 m y − z ∩ Z n = (mr − 1)P(u) + 1 m y ∩ Z n . The enumerative theory for systems of linear diophantine equations establishes that the function r → (mr − 1)P(u) + 1 m y ∩ Z n = (r − 1) mP(u) + (m − 1)P(u) + 1 m y ∩ Z n is a quasipolynomial; see Theorem 6.50 in [BG] or Theorem 4.6.11 in [Sta3]. Moreover, the leading coefficient of this quasi-polynomial is the relative volume of mP(u) with respect to the lattice α −1 (rmu) ∩ Z n ; see Theorem 6.55 in [BG]. Since mu ∈ ZA and (mr − 1)P(u) + 1 m y ⊂ α −1 (rmu), the normalized volume vol n−d (P(u)) equals (n − d)!/m n−d times coefficient of the degree n − d term in this quasi-polynomial, so By hypothesis, α is non-degenerate, so P(u) has dimension n −d if and only if u lies in the interior of Z. Therefore, we have It follows immediately from this polyhedral description that K A (t) inherits the symmetries of the zonotope Z. ✄ STEP 2. The second step takes advantage a well-known theorem from convex geometric analysis. To isolate the applicable result, recall that g : R d → R is log-concave if g(u) > 0 and log(g) is concave. Equivalently, for s ∈ [0, 1] and u, v ∈ Z d , we have the inequality g su Proposition 2.1. Let X ⊂ R n be a closed convex set and let π : R n → R d be a surjective linear map. If the function g : R d → R assigns to u ∈ R d the volume of the fibre π −1 (u) ∩ X , then g is log-concave.
Proof. Choosing a basis for ker(π) defines an injective linear map ϕ : R n−d → R n such that π • ϕ is the zero map. We may also choose an injective linear map η : R d → R n such that π • η is the identity map, because π is surjective. For any u ∈ R d , we set There exists a positive k ∈ R such that g(u) = k R n−d 1 X(u) (p)dp, so the Prékopa-Leindler inequality (Theorem 7.1 in [Gar] or Theorem 6.4 in [Vil]) yields To establish that the coefficient function of K A (t) is log-concave, we simply apply Proposition 2.1 when X = [0, 1] n and π = α, because Step 1 shows that the coefficients of K A (t) equal the normalized volume of the fibres of the map α : [0, 1] n → Z.
We finish this step by showing that the coefficient function of K A (t) is quasi-concave. A function g : R d → R is quasi-concave if, for all e ∈ R, its superlevel set {u ∈ R d : g(u) e} is convex. Equivalently, for s ∈ [0, 1] and u, v ∈ R d , we have g su + (1 − s)v min g(u), g (v) . In particular, every positive log-concave function is quasi-concave. ✄ STEP 3. Lastly, we reinterpret the coefficients of K A (t) as the normalized volumes of certain regions in the hypercube [0, m] n−d . This perspective is inspired by the well-known interpretation for the Eulerian numbers as the normalized volume of "slabs" in the hypercube; see [Sta1]. In fact, when A = [1 · · · 1], we recover this interpretation.
Without loss of generality, we may assume that A has a specified block structure. By reordering the columns and making a unimodular change of basis on Z d , it suffices to consider the case By definition, each region R(u) is a rational polytope and, combined together, these regions partition the hypercube [0, m] n−d . Hence, the union u R(u) has normalized volume equal to m n−d (n − d)!.
To complete the proof, it is enough to show that the coefficients of K A (t) also correspond to the normalized volumes of the regions R(u). If J is the integer (n × d)-matrix mH −1 0 , then we obtain AJ = mI d . Moreover, the inequalities m(u i − 1) b i · p mu i hold if and only if the inequalities 1 u i − 1 m b i · p 0 hold. It follows that the affine map p → 1 m (Ju − Bp) sends the region R(u) into the rational polytope P(u) = {x ∈ [0, 1] n : ∑ j x j a j = u} = α −1 (u) ∩ [0, 1] n . The inverse map is associated to the (n − d) × n)-matrix L := [ 0 | I n−d ] which satisfies LJ = 0 and LB = −mI n−d . Since Step 1 establishes that the coefficients of K A (t) equal the normalized volumes of the polytopes P(u), we conclude that K A (1) = m n−d (n − d)! as required. ✷ Remark 2.2. Suppose that A is a totally unimodular matrix which means m = 1. If u ∈ Z d , then P(u) is a lattice polytope (see Theorem 19.1 in [Sch]) and C r (u, v) = (r − 1)P(u) ∩ Z n . Thus, the normalized volume vol n−d P(u) is a nonnegative integer which implies that K A (t) ∈ Z[t ±1 ] and K A (1) = (n − d)!. Finding a combinatorial description for these coefficients remains an open problem. One tantalizing possibility is that the coefficients naturally enumerate some partition of the symmetric group on n − d letters.

OTHER CONNECTIONS
This section uses two additional examples to highlight some potential applications for Theorem 1.1.
To help orient future research, we also state a few open problems.
ASYMPTOTIC EQUALITIES. When F(1) = 0, Theorem 1.1 shows that some of the coefficients of Φ r [F(t)], regarded as functions of r, are asymptotically equal to an explicit multiple of r n−d . In many situations, Theorem 1.1 yields an asymptotic equality even though F(1) = 0. Roughly speaking, the Taylor expansion of F(t) about t = 1 allows one to cancel factors from the denominator and, thereby, apply Theorem 1.1 to a submatrix of A. We demonstrate this approach in the following prototypical example.
Under the standard grading (i.e. when A = [ 1 · · · 1 ]), the module M would be the homogeneous coordinate ring of the Veronese surface in P 5 . As a rational function, the Hilbert series of M is .
Since ℓ := codim(M) = 3 > 0, it follows that F(1) = 0. In this context, an appropriate expansion of F(t) comes from a choice of initial module. Each monomial initial module of M is homogeneous with respect to the Z n -grading arising from the identity matrix I n ; see [MS,§2.2]. Since M and any initial module have the same Hilbert series, the Taylor expansion for the numerator of the Z n -graded Hilbert series produces an expression for F(t) as a polynomial in the variables (1 − t a 1 ), . . ., (1 − t a n ); see Exercise 8.15 in [MS]. Moreover, the lowest terms in each such expression have degree ℓ, are square-free, and have nonnegative coefficients; see Exercise 8.8 in [MS]. For instance, the monomial initial ideals x 2 , x 3 , x 5 2 and x 1 , x 2 , x 2 5 ∩ x 2 2 , x 5 , x 6 of I (chosen from among the 29 possibilities) yield the following expansions: Observe that the coefficient of the term (1 − t a s 1 )(1 − t a s 2 )(1 − t a s 3 ) equals the multiplicity µ s of the minimal prime x s 1 , x s 2 , x s 3 for the initial ideal; cf. Definition 8.43 in [MS]. Since we have where A s is the (d × (n − ℓ))-submatrix of A in which the columns indexed by s are omitted and m A s is the greatest common divisor of the maximal minors of A s . In particular, we have lim r→∞ Φ r [F(t)] r = 1 2 t 1 t 2 (t 1 + 1)(t 2 + 1)(t 1 t 2 + 1)(1 − t 1 t 2 )(1 − t 2 1 t 2 )(1 − t 1 t 2 2 ) = 4 4 K 2 0 2 0 2 2 (t) (1−t a 2 )(1−t a 3 )(1−t a 5 ) = 2 2 K 2 0 2 1 2 2 (t) (1−t a 1 )(1−t a 2 )(1−t a 5 ) + 2 2 K 2 2 0 0 1 2 (t) (1−t a 2 )(1−t a 5 )(1−t a 6 ) .
As the expansions for F(t) vary, these equations lead to nontrivial relations among asymptotic K-polynomials associated to submatrices of A. ⋄ This approach also suggests a way to understand lower-order asymptotics. Specifically, if lim sup STOCHASTIC MATRICES. By scaling the matrix associated to the linear operator Φ r , we obtain a stochastic matrix. Specifically, the matrix C(r) := r d−n [C r (u, v) ] where u and v range over int(Z) ∩ Z d is a nonnegative square matrix each of whose columns sum to 1. When A = [ 1 · · · 1 ], [Hol] (also see [DF1]) establishes that C(r) has the following "amazing" properties: • for all r, the stationary vector (i.e. the eigenvector with eigenvalue 1) corresponds to the coefficients of K A (t)/(n − d)!; • the matrix C(r) has eigenvalues r −i for 0 i < n − d with explicit eigenvectors independent of r; and • we have C(r 1 )C(r 2 ) = C(r 1 r 2 ).
The next example illustrates how these properties extend to our more general setting.