Discrete Comput Geom (2008) 40: 365–376 DOI 10.1007/s00454-007-9002-5

Ehrhart Series and Triangulations

Sam Payne

Received: 13 February 2007 / Revised: 11 May 2007 / Published online: 18 September 2007 © Springer Science+Business Media, LLC 2007

Abstract We express the for lattice points in a rational poly- hedral cone with a simplicial subdivision in terms of multivariate analogues of the h- of the subdivision and “local contributions” of the links of its nonuni- modular faces. We also compute new examples of nonunimodal h∗-vectors of reflex- ive .

1Introduction

Let N be a lattice and let σ be a strongly convex rational polyhedral cone in N R = N R.Thegeneratingfunctionforlatticepointsinσ , ⊗Z v Gσ x , = v (σ N) ∈!∩ is a rational function, in the quotient field Q(N) of the multivariate Laurent polyno- mial ring Z N .Forinstance,ifσ is unimodular, spanned by a subset v1,...,vr of [ ] v v { } abasisforN,thenGσ is equal to 1/(1 x 1 ) (1 x r ). − ··· − Suppose " is a rational simplicial subdivision of σ ,withv1,...,vs the primi- tive generators of the rays of ".WedefineamultivariateanalogueH" of the h- dim τ dim σ dim τ h"(t) τ " t (1 t) − ,by = ∈ · − " vi vj H" x (1 x ) . = · − τ " # vi τ vj τ % !∈ $∈ $&∈

Supported by the Clay Institute. S. Payne (!) Department of Mathematics, Stanford University, Bldg. 380, 450 Serra Mall, Stanford, CA 94305, USA e-mail: [email protected] 366 Discrete Comput Geom (2008) 40: 365–376

Every point in σ is in the relative interior of a unique cone in " and can be written uniquely as a nonnegative integer linear combination of the primitive generators of the rays of that cone plus a fractional part. The generating function for lattice points in the relative interior of τ that have no fractional part is xvi /(1 xvi ),so vi τ the generating function for lattice points in σ with no fractional∈ part (with− respect v v & to the subdivision ")isH"/(1 x 1 ) (1 x s ).Inparticular,ifeveryconein " is unimodular, spanned by part− of a basis··· for− the lattice, then every lattice point has no fractional part, so this gives Gσ .Otherwise,theremaininglatticepointswith nonzero fractional part, which necessarily lie in the nonunimodular cones of ",may be accounted for as follows. Say that a cone is singular if it is not unimodular, and let "sing be the set of singular sing cones in ".Afterpossiblyrenumbering,sayτ " is spanned by v1,...,vr .Let Box(τ) be the open parallelepiped ∈ { }

Box(τ) a1v1 ar vr 0

v Bτ x . = v Box(τ) N ∈ ! ∩ We write lk τ for the link of τ in ".Inotherwords,lkτ is the union of the cones γ in " such that γ τ 0andγ τ is a cone in ".Wedefineamultivariateanalogue ∩ = + Hlk τ of the h-polynomial of lk τ by

vi vj Hlk τ x (1 x ) . = · − γ lk τ # vi γ vj (lk τ γ) % !∈ $∈ ∈ $! Theorem 1.1 Let " be a rational simplicial subdivision of a strongly convex rational polyhedral cone σ , and let v1,...,vs be the primitive generators of the rays of ". Then

v1 vs vi (1 x ) (1 x ) Gσ H" Bτ Hlk τ (1 x ) . − ··· − · = + · · − τ "sing # vi Star τ % ∈! &∈$ Here Star τ is the union of the maximal cones in " that contain τ . Recent work of Athanasiadis [1, 2], Bruns and Römer [9], and Ohsugi and Hibi [18]hashighlightedtheusefulnessofconsideringtheeffectsofa“specialsimplex” that is contained in all of the maximal faces of a triangulation. Our next result, which is inspired by their work, is a generalization of Theorem 1.1 that takes into account the effect of a special cone that is contained in all of the maximal cones of the subdivision. For a cone over a triangulated , the notion of special cone that we consider is slightly more general than a cone over a special simplex in the sense of [1]; a special cone is a cone over a special simplex if and only if it is not contained in the boundary of the cone over the polytope. Suppose, as above, that τ is a cone in " spanned by v1,...,vr ,andletλ be a face of τ .Afterpossiblyrenumbering,wemayassumethatλ is spanned by v1,...,vq , Discrete Comput Geom (2008) 40: 365–376 367 for some q r.WedefineapartiallyopenparallelepipedBox(τ,λ),whichwethink of as the Box≤ of τ relative to λ,as

Box(τ,λ) a1v1 ar vr 0 ai < 1foralli, and ai 0fori>q . ={ +···+ | ≤ =& } Let the polynomial Bτ,λ be the relative analogue of Bτ ,

v Bτ,λ x . = v Box(τ,λ) N ∈ ! ∩ Theorem 1.2 Let " be a rational simplicial subdivision of a strongly convex rational polyhedral cone σ , and let v1,...,vs be the primitive generators of the rays of ". Suppose λ " is contained in every maximal cone of ". Then ∈

v1 vs vj (1 x ) (1 x ) Gσ Hlk λ Bτ,λ Hlk τ (1 x ) . − ··· − · = + · · − τ λ # vj Star τ % !( &∈$

Since the zero cone is contained in every cone of " and Bτ,0 Bτ for all τ ,Theo- rem 1.1 is the special case of Theorem 1.2 where λ 0. = =

Remark 1.3 There are many ways of computing the rational function Gσ and its specializations, some of which are algorithmically efficient. The excellent survey ar- ticles [3]and[10]mayserveasintroductionstotheextensiveliteratureonthistopic. Efficient algorithms have been implemented in the computer program LattE [11]. Theorems 1.1 and 1.2,andtheirspecializationstolatticepolytopes(Corollaries3.1 and 3.3), seem to be useful in cases where it is especially easy to give subdivisions that are close to unimodular, and in studying families of such examples in which the contributions of the singular cones can be easily understood. See, for instance, the examples in Sect. 4,whichwerecomputedbyhand.

Theorems 1.1 and 1.2 specialize to give formulas for Ehrhart series of lattice poly- topes. See Sect. 3 for details. We use these specializations to construct examples of reflexive polytopes with interesting h∗-vectors, the results of which are summarized as follows. Suppose N is a lattice and P is a d-dimensional lattice polytope in N . ) R) Recall that the Ehrhart series of P is m EhrP (t) 1 # mP N ) t , = + { ∩ }· m 1 !≥ d 1 d and that (1 t) EhrP (t) h h t for some integers h .See[5,Chaps3 − + = 0∗ +···+ d∗ i∗ and 4] for these and other basic facts about Ehrhart series. We say that h∗(P ) i 1 = (h0∗,...,hd∗) is the h∗-vector of P ,andwritehP∗ (t) for the polynomial hi∗t . " 1There is unfortunately no standard notation for the numerator of the Ehrhart series of a lattice polytope and its coefficients, despite the extensive literature on the topic. The relation between this polynomial and the Ehrhart polynomial of a lattice polytope is analogous to the relation of the h-polynomial of a simplicial polytope with the f -polynomial; some like to call its vector of coefficients the “Ehrhart h-vector” [9]. Others simply denote it by h [2, 5, 18]orδ [12–15]. Here we use the notation h∗,followingStanley[20] and Athanasiadis [1], to emphasize the analogy with h-polynomials of simplicial polytopes while avoiding any possible ambiguity. 368 Discrete Comput Geom (2008) 40: 365–376

Recall that P is said to be reflexive if it contains 0 in its interior and its dual polytope has vertices in the dual lattice. Suppose that P is reflexive. Then hi∗ hd∗ i for all i [12]. Furthermore, if ∂P has a regular unimodular triangulation, given= by− intersecting ∂P with the domains of linearity of a convex piecewise linear function on N ,thenh (P ) is equal to the h-vector of this triangulation, which is combinatorially R) ∗ equivalent to the boundary complex of a simplicial polytope. In particular, if ∂P has aregularunimodulartriangulationthenh (P ) is unimodal, in the sense that h ∗ 0∗ ≤ h1∗ h∗d/2 ,andfurthermorethevectorofsuccessivedifferences ≤···≤ [ ]

g∗(P ) (h0∗,h1∗ h0∗,...,h∗d/2 h∗d/2 1) = − [ ] − [ ]− is a Macaulay vector, i.e. the Hilbert sequence of a graded algebra generated in degree one. For an arbitrary reflexive polytope, Hibi showed that h0∗ h1∗ hi∗ for 2 i

Theorem 1.4 For every d 6, there exists a d-dimensional reflexive simplex P such ≥ that h∗(P ) is not unimodal.

Theorem 1.5 For every d 4, there exists a d-dimensional reflexive simplex P such ≥ that h∗(P ) is unimodal, but g∗(P ) is not a Macaulay vector.

Theorem 1.6 For any positive integers m and n, there exists a reflexive polytope P and indices i1

Remark 1.7 None of the examples of reflexive polytopes with nonunimodal h∗- vectors constructed here are normal, in the sense where a lattice polytope P is normal if every lattice point in mP is a sum of m lattice points in P ,forallpositiveintegers m.FornormalreflexivepolytopesP ,thequestionsofwhetherh∗(P ) is unimodal and whether g∗(P ) is a Macaulay vector remain open and interesting [18].

We conclude the introduction with an example illustrating Theorem 1.2.

Example 1.8 Suppose N Z3, v (0, 0, 1) and = =

v1 (1, 0, 1), v2 (0, 1, 1), v3 (0, 1, 1), v4 ( 1, 0, 1). = = = − = − Discrete Comput Geom (2008) 40: 365–376 369

Let σ be the cone spanned by v1,v2,v3, and v4,with" the simplicial subdivision of σ whose maximal cones are

σ1 v1,v2,v3 and σ2 v2,v3,v4 . =+ , =+ , Then σ1 and σ2 are not unimodular; the lattice points with no fractional part are exactly those (a, b, c) such that a b c is even. The remaining lattice points may be written uniquely as v plus a lattice+ + point with no fractional part. We compute the generating function Gσ as follows, using Theorem 1.2 with λ v2,v3 . =+ , There are exactly three cones in lk λ:0, v1 ,and v4 . Therefore + , + , v1 v4 v1 v4 v4 v1 Hlk λ (1 x )(1 x ) x (1 x ) x (1 x ), = − − + − + − 1 xv1 xv4 . = −

Since Boxλ,λ contains a unique lattice point v and Boxσ1,λ and Boxσ2,λ contain no lattice points, it follows that

(1 xv)(1 xv1 xv4 ) Gσ + − . = (1 xv1 )(1 xv2 )(1 xv3 )(1 xv4 ) − − − −

2ProofofTheorems1.1 and 1.2

As observed in the introduction, Theorem 1.1 is a special case of Theorem 1.2.We will begin by showing the converse, that Theorem 1.2 is a consequence of Theo- rem 1.1,usingthefollowinglemma.Thelemmaisamultivariateanalogueofthe familiar fact that, for any simplicial complex "),theh-polynomial of the join of ") with a simplex is equal to the h-polynomial of ").

Lemma 2.1 If λ is a face of every maximal cone of " then H" Hlk λ and = Hlk γ Hlk(γ λ), = + for every γ ". ∈

Proof We show that H" Hlk λ.Theproofofthesecondclaimissimilar.Every maximal face of " contains=λ if and only if " is the join of λ with lk λ.Sinceλ is the join of its rays, it will suffice to consider the case where λ is one-dimensional, with primitive generator v1.Inthiscase,therequiredidentitymaybeseenbyregrouping the terms in the summation defining H" as

vi vj vk v( H" x (1 x ) x (1 x ) . = · − + · − v1 γ # vi γ vj γ % v1 τ # vk τ v( τ % !&∈ $∈ $&∈ !∈ $∈ $&∈ Since the cones not containing v1 are exactly the γ lk λ,andsincetheconescon- ∈ taining v1 are exactly those τ (γ λ) for γ lk λ,theaboveequationgives = + ∈ v1 v1 H" (1 x ) Hlk λ x Hlk λ, = − · + · so H" Hlk λ,asrequired. = ! 370 Discrete Comput Geom (2008) 40: 365–376

Suppose λ is contained in every maximal cone. Then, for any cone τ containing λ,

Bτ,λ Bγ . = (γ λ) τ +!=

Futhermore, for each γ such that (γ λ) τ ,wehaveStarγ Star τ ,andHlk γ + = = = Hlk τ ,byLemma2.1.Therefore,Theorem1.2 follows from Theorem 1.1,whichwe now prove.

Proof of Theorem 1.1 Let v be a lattice point in σ .Thenv is contained in the relative interior of a unique cone γ ".Ifv1,...,vr are the primitive generators of the rays of γ ,thenv can be written uniquely∈ as

v a1v1 ar vr v , = +···+ +{ } where each ai is a nonnegative integer and v ,whichwecallthefractionalpartof v,iseitherzeroorliesinBox(τ) for some{ unique} singular cone τ γ .Sincev - lies in the relative interior of γ , ai must be strictly positive for each vi in (γ ! τ). Conversely, if v is a lattice point in Box(τ),andv a1v1 ar vr v ,where ) = +···+ + ) the ai are nonnegative integers that are strictly positive for vi (γ ! τ),thenv is in the relative interior of γ and v v .Therefore, ∈ { }= ) v x ) Hlk τ · (1 xvi ) vi Star τ ∈ − & is the generating function for lattice points v σ such that v v).ThenBτ H / (1 xvi ) is the generating function∈ for lattice points{ }= in σ whose frac-· lk τ vi Star τ ∈ − tional part is in Box(τ),andthetheoremfollows. ! &

3SpecializationtoLatticeTriangulationsofPolytopes

Let N be a lattice, and let P be a d-dimensional lattice polytope in N .Suppose ) R) N N ) Z,withσ the cone over P 1 in NR,andletu N Z be the projection to the= second× factor. Since ×{ } : →

# mP N ) # v (σ N) u(v) m { ∩ }= { ∈ ∩ | = } for all positive integers m,thespecialization

v u(v) ϕ Q(N) Q(t), x t : → 0→ maps Gσ to EhrP (t). Suppose is a lattice triangulation of P ,andlet" be the subdivision of σ con- T sisting of the cones τF over F 1 for all faces F .Theneachoftheprimitive ×{ } ∈ T generators v1,...,vs of the rays of " is a lattice point in P 1 ,sou(vi) 1for all i.Itfollowsthat ×{ } = (s d 1) ϕ(H") (1 t) − − h (t), = − · T Discrete Comput Geom (2008) 40: 365–376 371 where h (t) is the h-polynomial of the simplicial complex .Similarly, T T (s d 1) ϕ(Hlk τ ) (1 t) )− )− hlk F (t), F = − · sing where s) is the number of vertices in lk F ,andd) dim lk F .Let denote the set sing = T of nonunimodular simplices of .ForF ,letBF (t) ϕ(Bτ ),so T ∈ T = F u(v) BF (t) t . = v Box(τF ) N ∈ ! ∩ The following specialization of Theorem 1.1 is due to Betke and McMullen [7,The- orem 1] and was rediscovered by Batyrev and Dais in the context of stringy algebraic geometry [4,Theorem6.10].

Corollary 3.1 Let P be a lattice polytope, and let be a lattice triangulation of P . Then T

hP∗ (t) h (t) BF (t) hlk F (t). = T + · F sing ∈!T

Note that BF (t) and hlk F (t) have nonnegative integer coefficients, so the theorem of Stanley that hP∗ (t) has nonnegative integer coefficients [19,Theorem2.1]follows immediately. Another interesting proof of this nonnegativity, using “irrational de- compositions,” recently appeared in work of Beck and Sottile [6]. Furthermore, hlk F (t) is always nonzero, and BF (t) must be nonzero for some F if is not unimodular. Therefore, from Corollary 3.1,wededucethefollowing. T Corollary 3.2 [7,Theorem2] Let P be a lattice polytope, and let be a lattice triangulation of P . Then T

h∗(P ) hi( ) i ≥ T for all i. Furthermore, equality holds for all i if and only if is unimodular. T

If the triangulation contains a special simplex F ) that is a face of every maximal simplex of ,thenwecantakethisintoaccountusingTheoremT 1.2.ForF F ,let T ≥ ) BF,F (t) ϕ(BτF ,τ ). ) = F ) Corollary 3.3 Let P be a lattice polytope, and let be a triangulation of P with a T special simplex F ). Then

h∗ (t) h F (t) BF,F (t) hlk F (t). P = lk ) + ) · F F !≥ ) Recall that a lattice polytope P is called reflexive if it contains 0 in its interior and the polar dual polytope of P has vertices in the dual lattice of N ).Inthespecial case where P is reflexive and 0 F ) is a special simplex of the triangulation ,then is the join of 0 with a triangulation= of the boundary of P ,andwerecover[T 16, TTheorem 1.3]. { } 372 Discrete Comput Geom (2008) 40: 365–376

Remark 3.4 In [9], Bruns and Römer use techniques from commutative algebra to show that if mP is a translate of a reflexive polytope for some positive integer m, and if P has any regular unimodular triangulation, then P has a regular unimodular triangulation with an (m 1)-dimensional special simplex F ) such that lk F ) is com- binatorially equivalent to− the boundary complex of a simplicial polytope. From this they deduce that h (P ) h(lk F ) is unimodal and that g (P ) is a Macaulay vector. ∗ = ) ∗ Remark 3.5 These specializations to lattice polytopes are closely related to stringy invariants of toric varieties. See [4]and[16]fordetailsonthisconnection.Special simplices have also appeared in the stringy geometry literature, where they have been called the “core” of a triangulation [21].

4ExamplesofReflexivePolytopeswithNonunimodalh∗-Vectors

Let N ) be a lattice and let M) Hom(N ), Z) be its dual lattice. Let P be a d- dimensional lattice polytope in N=.RecallthatP is reflexive if and only if it contains R) 0initsinteriorandthedualpolytope

P ◦ u M) u, v 1forallv P ={ ∈ R |+ ,≥− ∈ } has vertices in M).

Proposition 4.1 Let a1,...,ad , and b be positive integers, and let 1 f (a1,...,ad ). = b

d Then the simplex P conv e1,...,ed , f in R is reflexive with respect to the d = { − } lattice N ) Z Z f if and only if a1 ad bc for some integer c and each = + · +···+ = ai divides b(c 1). + d Proof Let u e∗ e∗.ThenP ◦ is the simplex in (R )∗ with vertices u and =− 1 −···− d

a1 ad b ui u +···+ + ei∗, = + ai for 1 i d.Nowu lies in M if and only if u, f (a1 ad )/b is some ≤ ≤ ) + ,= +···+ integer c.Ifthisisthecase,thenui lies in M if and only if ui,ei b(c 1)/ai 1 ) + ,= + − is an integer. !

In the following examples, we use Proposition 4.1 to construct a reflexive simplex P ,andthenapplyCorollary3.3 with respect to the triangulation obtained by taking the join of 0 with the boundary of P ,andthespecialsimplexTF 0, to compute { } ) = h∗(P ).Thelinkofa(d r)-dimensional face F that contains 0 is the boundary − ∈ T r complex of an r simplex, so hlk∗ F (t) 1 t t . We write v (v, 1) for the d =d + +···+ = lift of a point v R to height one in R R. ∈ × ' Discrete Comput Geom (2008) 40: 365–376 373

Example 4.2 Suppose d bk r for some positive integers b and k and some non- = + d negative integer r.Letf be the following vector in R , 1 f (1,...,1,b,...,b). = b bk r

By Proposition 4.1, P conv e1,...,e( )*d , +f ( is)* reflexive+ with respect to the lattice d = { − } N ) Z Z f .TheonlynonunimodularfaceF of such that Box(τF , 0) con- = + · T tains lattice points is F conv 0,e1,...,ebk .Letf ) f ebk 1 ed .Then = { } = − + −···− Box(F, 0) (N ) Z) consists of the lattice points ∩ × 1 2 b 1 (e1 ebk), (e1 ebk), . . . , − (e1 ebk), b +···+ b +···+ b +···+ which have final' coordinates' k,'2k,...,(b' 1)k,respectively.Therefore' ' BF,0(t) tk t(b 1)k,and − = +···+ − d r k (b 1)k h∗ (t) (1 t ) (1 t )(t t − ). P = +···+ + +···+ +···+ Example 4.3 Suppose d 7, and = 1 f (1, 2, 2, 4, 4, 4, 4). = 7

By Proposition 4.1, P conv e1,...,e7, f is reflexive with respect to the lat- 7 = { − } tice N ) Z Z f .LetF1, F2,andF3 be the faces of with vertex sets = + · T 0,e1,...,e7 , 0,e1,e2,e3, f ,and 0,e1, f ,respectively.Thenthelattice { } { − } { − } points in Box(F1, 0) (N ) Z) are ∩ ×

(1/7)(e1 2e2 2e3 4e4 4e5 4e6 4e7), + + + + + + (1/7)(2e1 4e2 4e3 e4 e5 e6 e7), ' +' +' +' + '+ '+ ' (1/7)(3e1 6e2 6e3 5e4 5e5 5e6 5e7), ' + ' + ' +' +' '+ '+ (1/7)(4e1 e2 e3 2e4 2e5 2e6 2e7), ' + + + + + + (1/7)(5e1 3'e2 3'e3 6'e4 6'e5 6'e6 6'e7), + + + + + + (1/7)(6'e1 '5e2 '5e3 '3e4 '3e5 '3e6 '3e7), + + + + + + ' ' ' ' ' ' ' 2 which have final coordinates' 3,' 2, 5,' 2, 5, and' 4,' respectively.' ' So BF1,0(t) 2t 3 4 5 = + t t 2t .Similarly,thelatticepointsinBox(F2, 0) (N ) Z) are + + ∩ ×

(1/4)( 3f e1 2e2 2e3), − + + + (1/4)( f 3e1 2e2 2e3), − + + + ' ' ' ' 2 which both have final coordinate 2, so' BF ,0(t) 2t .Finally,theuniquelatticepoint '2 =' ' in Box(F3, 0) (N ) Z) is (1/2)( f e1),soBF3,0(t) t. For all other∩ faces×F ,Box(F, 0)−contains+ no lattice points.= It follows that ' h∗(P ) (1, 2, 6, 5, 5, 6, 2, 1). = 374 Discrete Comput Geom (2008) 40: 365–376

Example 4.4 Suppose d 11, and = 1 f (1, 1, 1, 2, 4, 4, 4, 4, 4, 4, 4). = 11 11 By Proposition 4.1, P is reflexive with respect to the lattice N ) Z Z f .LetF1, = + · F2,andF3 be the faces of with vertex sets 0,e1,...,e11 , 0,e1,e2,e3,e4, f , T { } { − } and 0,e1,e2,e3, f ,respectively.Then,bycomputationssimilartothoseinEx- ample{ 4.3, − }

2 3 5 6 8 9 BF ,0(t) t 2t 2t 2t 2t t , 1 = + + + + + 2 3 BF ,0(t) t t , 2 = + 2 BF ,0(t) t , 3 = and BF,0(t) 0forallotherF .Itfollowsthat = h∗(P ) (1, 1, 4, 6, 4, 6, 6, 4, 6, 4, 1, 1). =

Proof of Theorem 1.4 If b 3andk>r 1inExample4.2,thenh∗(P ) is not unimodal. In particular, taking≥ b 3andr +0, 1, or 2, produces reflexive simplices = = with nonunimodal h∗-vectors in all dimensions d 6exceptford 7, 8, or 11. For d 8, one may take b 4andk 2. For dimensions≥ 7 and 11, reflexive= simplices = = = with nonunimodal h∗-vectors are given by Examples 4.3 and 4.4,respectively. !

Proof of Theorem 1.5 If b 2andk 2inExample4.2,thenh (P ) is unimodal, but = ≥ ∗ h1∗ h0∗ 0andhk∗ hk∗ 1 1, so g∗(P ) is not a Macaulay vector. Taking b k 2 produces− = such examples− − in= all dimensions d 4. = = ≥ ! One key ingredient in the proof of Theorem 1.6 is the following special case of Braun’s formula [8], which gives the h∗-polynomial of the free sum of two reflexive polytopes. Recall that if Q and Q) are polytopes in vector spaces V and V ),respec- tively, each containing 0 in its interior, then the free sum Q Q) is the of Q 0 and 0 Q in V V . ⊕ ×{ } { }× ) × )

Braun’s Formula Let Q and Q) be reflexive polytopes. Then

hQ∗ Q (t) hQ∗ (t) hQ∗ (t). ⊕ ) = · ) Note that the dual of the free sum of two polytopes is the product of their respective duals, so the free sum of two reflexive polytopes is reflexive.

Proof of Theorem 1.6 Nill has constructed a sequence of reflexive simplices Qj such that dim Qj j and the normalized volume = vol Qj h∗(Qj ) h∗(Qj ) = 0 +···+ j grows doubly exponentially with j [17]. Therefore, there exists a reflexive simplex Q such that h (Q) n for some i,anddim(Q) O(log log n).LetQ be the reflexive i∗ ≥ = ) Discrete Comput Geom (2008) 40: 365–376 375 simplex constructed by taking b m 1, k dim(Q) 2, and d bk in Exam- = + = + = ple 4.2.Thendim(Q)) (m 1)(dim(Q) 2),sodim(Q Q)) O(mlog log n), and = + + ⊕ = d k 2k (b 1)k h∗ (t) (1 t t ) (t t t − ). Q) = + +···+ + + +···+ Let P Q Q .ThenP is reflexive and, by Braun’s Formula, h (t) h (t) = ⊕ ) P∗ = Q∗ · h∗ (t).Itfollowseasilythat Q)

hk∗( i(P ) vol(Q) hi∗(Q), + = + for 1 ( m 1, and ≤ ≤ + hk(∗ ( 1) 1(P ) vol(Q), + − = for 1 ( m.Inparticular,settingi( k( i and j( k(( 1) 1, we have h ≤h ≤n and h h n,for1 =( +m,asrequired.= + − i∗( j∗( i∗( 1 j∗( ! − ≥ + − ≥ ≤ ≤ Acknowledgements This note was written in response to questions raised at the 2006 Snowbird confer- ence on Integer Points in Polyhedra. I am grateful to the organizers of the conference for the opportunity to participate and thank A. Barvinok, M. Beck, B. Braun, J. De Loera, C. Haase, T. Hibi, and S. Ho¸sten for stimulating conversations. I also thank P. McMullen and the referee for helpful comments and corrections.

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