Inequalities for f*-vectors of lattice polytopes

Matthias Beck; Danai Deligeorgaki; Max Hlavacek; Jerónimo Valencia-Porras

doi:10.1515/advgeom-2024-0002

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Inequalities for f^*-vectors of lattice polytopes

Matthias Beck , Danai Deligeorgaki , Max Hlavacek and Jerónimo Valencia-Porras

Published/Copyright: April 26, 2024

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From the journal Advances in Geometry Volume 24 Issue 2

Abstract

The Ehrhart polynomial ehr_P(n) of a lattice polytope P counts the number of integer points in the n-th dilate of P. The f^*-vector of P, introduced by Felix Breuer in 2012, is the vector of coefficients of ehr_P(n) with respect to the binomial coefficient basis {n−10,n−11,…,n−1d}, where d = dim P. Similarly to h/h^*-vectors, the f^*-vector of P coincides with the f-vector of its unimodular triangulations (if they exist). We present several inequalities that hold among the coefficients of f^*-vectors of lattice polytopes. These inequalities resemble striking similarities with existing inequalities for the coefficients of f-vectors of simplicial polytopes; e.g., the first half of the f^*-coefficients increases and the last quarter decreases. Even though f^*-vectors of polytopes are not always unimodal, there are several families of polytopes that carry the unimodality property. We also show that for any polytope with a given Ehrhart h^*-vector, there is a polytope with the same h^*-vector whose f^*-vector is unimodal.

Keywords: Lattice polytope; Ehrhart polynomial; Gorenstein polytope; f*-vector; h*-vector; unimodality

MSC 2010: Primary 52B20; Secondary 05A15; 52C07

1 Introduction

For a d-dimensional lattice polytope P ⊂ ℝ^d (i.e., the convex hull of finitely many points in ℤ^d) and a positive integer n, let ehr_P(n) denote the number of integer lattice points in nP. Ehrhart’s famous theorem [11] says that ehr_P(n) evaluates to a polynomial in n. Similar to the situations with other combinatorial polynomials, it is useful to express ehr_P(n) in different bases; here we consider two such bases consisting of binomial coefficients:

ehrP⁡(n)=∑k=0dhk∗n+d−kd=∑k=0dfk∗n−1k. (1)

We call (f0∗,f1∗,…,fd∗) the f^*-vector and (h0∗,h1∗,…,hd∗) the h^*-vector of P. Stanley [15] proved that the h^∗-vector of any lattice polytope is nonnegative (whereas the coefficients of ehr_P(n) written in the standard monomial basis can be negative). Breuer [9] proved that the f^*-vector of any lattice polytopal complex is nonnegative (whereas the h^∗-vector of a complex can have negative coefficients); his motivation was that various combinatorially-defined polynomials can be realized as Ehrhart polynomials of complexes and so the nonnegativity of the f^*-vector yields a strong constraint for these polynomials. The f^*- and h^*-vector can also be defined through the Ehrhart series of P:

EhrP⁡(z):=1+∑n≥1ehrP⁡(n)zn=∑k=0dhk∗zk(1−z)d+1=1+∑k=0dfk∗(z1−z)k+1.

It is thus sometimes useful to add the definition f−1∗ := 1. The polynomial ∑k=0dhk∗zk is the h^*-polynomial of P, and its degree is the degree of P. For general background on Ehrhart theory, see, e.g., [3]. The f^*- and h^*-vectors share the same relation as f- and h-vectors of polytopes/polyhedral complexes, namely

∑k=0dhk∗zk=∑k=0d+1fk−1∗zk(1−z)d−k+1 (2)

hk∗=∑j=−1k−1(−1)k−j−1d−jk−j−1fj∗ (3)

fk∗=∑j=0k+1d−j+1k−j+1hj∗. (4)

The (very special) case that P admits a unimodular triangulation yields the strongest connection between f^*/h^*-vectors and f/h-vectors: in this case the f^*/h^*-vector of P equals the f/h-vector of the triangulation, respectively.

Example 1

Let P be the square [−1, 1]². The unimodular triangulation of P shown in Figure 1 has f-vector (f₀, f₁, f₂) = (9, 16, 8), where f_i counts its i-dimensional faces. Equivalently, f^*(P) = (9, 16, 8), and one easily checks that (1) yields the familiar Ehrhart polynomial ehr_P(n) = (2n + 1)².

Figure 1

A (regular) unimodular triangulation of the square [−1, 1]².

Example 2

The f^*-vector of a d-dimensional unimodular simplex Δ equals [d+11,d+12,…,d+1d+1], coinciding with the f-vector of Δ considered as a simplicial complex. If we include f−1∗ = 1, it gives the only instance of a symmetric f^*-vector of a lattice polytope P, since the equality f−1∗ = fd∗ combined with (4) implies that hi∗ = 0 for 1 ≤ i ≤ d.

There has been much research on (typically linear) constraints for the h^*-vector of a given lattice polytope; see, e.g., [16; 17]. On the other hand, f^*-vectors seem to be much less studied, and our goal is to rectify that situation. Our motivating question is how close the f^*-vector of a given lattice polytope is to being unimodal, i.e., the f^*-coefficients increase up to some point and then decrease. Our main results are as follows.

Theorem 3

Let d ≥ 2 and let P be a d-dimensional lattice polytope. Then

f0∗<f1∗<⋯<fd2−1∗≤fd2∗;
f3d4∗>f3d4+1∗>⋯>fd∗;
fk∗≤fd−1−k∗ for integer 0≤k≤d−22.

Examples 1 and 2 yield cases of polytopes for which the inequalities f⌊3d4⌋−1∗<f⌊3d4⌋∗ and f⌊d2⌋∗>f⌊d2⌋+1∗ hold, respectively, and thus the strings of inequalities in (a) and (b) can, in general, not be extended further. We record the following immediate consequence of Theorem 3.

Corollary 4

Let P be a d-dimensional lattice polytope. Then for 0 ≤ k ≤ d,

fk∗≥min{f0∗,fd∗}.

We remark that one can prove that if P is of degree d ≥ 2, equal to its dimension, then f0∗≤fd∗, except when the h^*-vector of P satisfies h2∗=⋯=hd∗=1.

Theorem 5

The f^*-vector of a d-dimensional lattice polytope, where 1 ≤ d ≤ 13, is unimodal. On the other hand, there exists a 15-dimensional lattice simplex with nonunimodal f^*-vector.

Even though f^*-vectors are quite different from f-vectors of polytopes, the above results resemble striking similarities with existing theorems on f-vectors. Namely, Björner [5; 6; 7] proved that the f-vector of a simplicial d-polytope satisfies the inequalities in Theorem 3 (with the ^*s removed, the last coordinate dropped, and d−22 replaced by d−32 in (c)). In fact, Björner also showed that in the f-analogue of Theorem 3(b) the decrease starts from ⌊3(d−1)4⌋ instead of ⌊3d4⌋ , and that the inequalities in Theorem 3(a) and (b) cannot be further extended, by constructing a simplicial polytope with f-vector that peaks at f_j, for any j with ⌊d2⌋≤j≤⌊3(d−1)4⌋.

Corollary 4 compares the entries of the f^*-vector with the minimum between the first and the last entry. Note that a similar relation for f-vectors of polytopes was recently proven by Hinman [14], answering a question of Bárány from the 1990s. (Hinman also proved a stronger result, namely certain lower bounds for the ratios f_k/f₀ and f_k/f_d−1.)

The f-analogue of Theorem 5 for simplicial polytopes is again older: Björner [5] showed that the f-vector of any simplicial d-polytope is unimodal for d ≤ 15 (later improved to d ≤ 19 by Eckhoff [10]), and he and Lee [4] produced examples of 20-dimensional simplicial polytopes with nonunimodal f-vectors.

For a special class of polytopes we can increase the range in Theorem 3(b). A lattice polytope P is Gorenstein of index g if

nP contains no interior lattice points for 1 ≤ n < g,
gP contains a unique interior lattice point, and
ehr_P (n − g) equals the number of interior lattice points in nP, for n > g.

This is equivalent to P having degree d + 1 − g and a symmetric h^*-vector (with respect to its degree).

Theorem 6

Let P be a d-dimensional Gorenstein polytope of index g. Then

fk−1∗>fk∗for12d+1+d+1−g2≤k≤d.

Going even further, for a certain class of polytopes we can prove unimodality of the f^*-vector, a consequence of the following refinement of Theorem 3(b) for polytopes with degree < d2 .

Theorem 7

Let P be a d-dimensional lattice polytope with positive degree ≤ s. Then

fk−1∗>fk∗for⌈d+s2⌉≤k≤d,

unless the degree of P is 0, i.e., P is a unimodular simplex with f^*-vector as in Example 2.

This theorem implies that lattice d-polytopes of degree s satisfying s² − s − 1 ≤ d2 have a unimodal f^*-vector (see Proposition 9 below for details). One family with asymptotically small degree, compared to the dimension, is given by taking iterated pyramids. Given a polytope P ⊂ ℝ^d, we denote by Pyr(P) ⊂ ℝ^d+1 the convex hull of P and the (d + 1)st unit vector. It is well known that P and Pyr(P) have the same h^*-vector (ignoring an extra 0), and so we conclude:

Corollary 8

If P is any lattice polytope then Pyrⁿ(P) has unimodal f^*-vector for sufficiently large n.

2 Proofs

We start with a few warm-up proofs which only use the fact that h^*-vectors are nonnegative.

Proof of Theorem 3(a)

It follows by (4) and the nonnegativity of h^*(P) that, for 1 ≤ k ≤ d2 ,

fk∗−fk−1∗=∑j=0k+1d+1−jk+1−j−d+1−jk−jhj∗≥0.

In fact, fk∗−fk−1∗ is bounded below by d+1k+1−d+1kh0∗>0, for 1 ≤ k < d2 , since h0∗ = 1. □

Proof of Theorem 3(c)

For 0 ≤ k ≤ d−22 , it follows by (4) that

fd−1−k∗−fk∗=∑j=0d−kd+1−jd−k−j−d+1−jk+1−jhj∗=∑j=0d−1−2kd+1−jk+1−d+1−jk+1−jhj∗+∑j=d−2kd−kd+1−jd−k−j−d+1−jk+1−jhj∗.

We have d+1−jk+1−d+1−jk+1−j≥0 since k+1−j≤k+1≤d+1−j2 holds for 0 ≤ j ≤ d − 1 − 2k. Similarly, d+1−jd−k−j−d+1−jk+1−j≥0 holds because k+1−j≤d−k−j≤d+1−j2 for d − 2k ≤ j. Therefore, it follows by the nonnegativity of the h^*-vector that fd−1−k∗−fk∗≥0. □

Proof of Theorem 7

Since hj∗ = 0 for j ≥ s + 1, equation (4) gives

fk−1∗−fk∗=∑j=0sd+1−jk−j−d+1−jk+1−jhj∗=∑j=0s2k−d−jk+1−jd+1−jk−jhj∗.

For d+s2≤k≤d, we have k + 1 − j > 0 and 2k − d − j > 0 for all j = 0, …, s − 1, and k + 1 − j > 0, 2k − d − j ≥ 0 for j = s. Therefore, the claim follows by the nonnegativity of the h^*-vector and the positivity of h0∗ . □

Proposition 9

Let P be a d-dimensional lattice polytope that has degree at most s, with s ≥ 1. If d ≥ 2s² − 2s − 2 then the f^*-vector of P is unimodal with a (not necessarily “sharp”) peak at fp∗ , where ⌊d2⌋≤p≤⌈d+s2⌉−1.

Proof

By Theorems 3(a) and 7, it suffices to show that f⌊d2⌋+i∗≥f⌊d2⌋+i+1∗ implies f⌊d2⌋+i+1∗≥f⌊d2⌋+i+2∗, or that 2f⌊d2⌋+1+i∗−f⌊d2⌋+2+i∗−f⌊d2⌋+i∗≥0 for 0 ≤ i ≤ s2 − 2.

As hj∗ = 0 for j ≥ s + 1, by equation (4) we can express 2f⌊d2⌋+1+i∗−f⌊d2⌋+2+i∗−f⌊d2⌋+i∗ as the sum

∑j=0s2d+1−j⌊d2⌋+2−j+i−d+1−j⌊d2⌋+3−j+i−d+1−j⌊d2⌋+1−j+ihj∗ =∑j=0s2⌈d2⌉−i⌊d2⌋+2−j+i−(⌈d2⌉−i)(⌈d2⌉−1−i)(⌊d2⌋+2−j+i)(⌊d2⌋+3−j+i)−1d+1−j⌊d2⌋+1−j+ihj∗.

Since d ≥ max{2s² − 2s − 2, 0} we have that (⌊d2⌋+3−j+i)(⌊d2⌋+2−j+i) is positive for j = 0, …, s and since h^* is nonnegative, it remains to show that

2(⌈d2⌉−i)(⌊d2⌋+3−j+i)−(⌈d2⌉−i)(⌈d2⌉−1−i)−(⌊d2⌋+2−j+i)(⌊d2⌋+3−j+i) =d−(2j−5)(⌈d2⌉−⌊d2⌋)+4i(⌈d2⌉−⌊d2⌋)−12i+4ij−4i2−6+5j−j2 =d−4i2+4ij−12i−j2+5j−6 if d is even,d−4i2+4ij−8i−j2+3j−1 if d is odd, (5)

is nonnegative for 0 ≤ j ≤ s. Indeed, the conditions j ≤ s and 0 ≤ i ≤ s2 − 2 imply that (5) is bounded below by

d−4i2−12i−j2−6≥d−4(s2−2)2−12(s2−2)−s2−6=d−2s2+2s+2,

which is nonnegative by assumption. □

The next proofs use more than just the nonnegativity of the h^*-vector. The first result needs the following elementary lemma on binomial coefficients.

Lemma 10

Let j, k, n be positive integers such that k ≤ n + 1 − j. Then, for n ≠ 2k − 1,

nk−nk−1≥n−jk−n−jk−1.

Proof

It suffices to prove the statement for the cases i) j = 1 and the quantities nk−nk−1 and n−1k−n−1k−1 having the same sign, and ii) the point when the signs change, i.e., n = 2k and j = 2. To show case i), we simplify

nk−nk−1=(n−1)!k!(n−k)!nn−k+1n−2k+1

and

n−1k−n−1k−1=(n−1)!k!(n−k)!n−2k.

If n ≥ 2k then the inequalities

nn−(k−1)(n−2k+1)≥n−2k+1>n−2k

imply that

nk−nk−1>n−1k−n−1k−1. (6)

If n ≤ 2k − 2, we have k(− 2k + 2 + n) ≤ 0 which is equivalent to

nn−(k−1)(2k−n−1)≥2k−n

and so again (6) holds as a weak inequality.

To show case ii), we compute

2kk−2kk−1=(2k)!k!(k+1)!=(2k−2)!k!(k−1)!2k(2k−1)k(k+1)

and

2k−2k−2k−2k−1=(2k−2)!k!(k−1)!.

Since 2(2k − 1) ≥ (k + 1) for any positive k, we conclude that

2kk−2kk−1≥2k−2k−2k−2k−1.

□

We are now prepared to prove Theorem 3(b).

Proof of Theorem 3(b)

The inequality fd−1∗>fd∗ holds by Theorem 7. Now, let 3d4+1≤k<d. By equation (4),

fk−1∗−fk∗=∑j=0k+1d+1−jk−j−d+1−jk+1−jhj∗. (7)

The difference d+1−jk−j−d+1−jk+1−j is nonnegative whenever k−j≥⌊d+1−j2⌋ and negative otherwise. Hence, the difference is nonnegative whenever j ≤ 2k − d and negative whenever j > 2k − d. Since 2d − 2k < 2k + 1 − d for ⌊3d4⌋ +1 ≤ k, from (7) we obtain

fk−1∗−fk∗≥∑j=02d−2kd+1−jk−j−d+1−jk+1−jhj∗ (8)

+∑j=2k+1−dk+1d+1−jk−j−d+1−jk+1−jhj∗ (9)

where the differences appearing in (8) are nonnegative and the ones in (9) are negative. Our aim is to compare the sums in (8) and (9) so as to conclude that fk−1∗−fk∗ is positive.

Using standard identities for binomial coefficients, the right hand-side of (8) equals

∑j=02d−2k∑l=j2d−2k−1d−lk−l−d−lk+1−l+2k−d+13k−2d−2k−d+13k−2d+1hj∗=∑l=02d−2k−1d−lk−l−d−lk+1−l∑j=02d−2k−1−lhj∗+2k−d+13k−2d−2k−d+13k−2d+1∑j=02d−2khj∗,

whence we conclude that the right hand-side of (8) is bounded below by

dk−dk+1h0∗+2k−d+13k−2d−2k−d+13k−2d+1∑j=02d−2khj∗>2k−d+13k−2d−2k−d+13k−2d+1∑j=02d−2khj∗ (10)

since dk−dk+1>0 for 3d4+1≤k<d, and h0∗=1,hj∗≥0 for j = 1, …, 2d − 2k − 1.

On the other hand, for the differences appearing in (9), using that 2d − 2k < j and j ≤ k + 1, it follows by Lemma 10 that

d+1−(2d−2k)d+1−k−d+1−(2d−2k)d−k≥d+1−jd+1−k−d+1−jd−k,

i.e.,

2k−d+13k−2d−2k−d+13k−2d+1≥d+1−jk−j−d+1−jk+1−j.

Hence for j ≥ 2k + 1 − d,

−2k−d+13k−2d−2k−d+13k−2d+1≤d+1−jk−j−d+1−jk+1−j.

Since both −d+1−jk−j+d+1−jk+1−j and hj∗ are nonnegative for j ≥ 2k + 1 − d, the sum in (9) is bounded below by

−2k−d+13k−2d−2k−d+13k−2d+1∑j=2k+1−ddhj∗. (11)

Now (10) and (11) yield

fk−1∗−fk∗>2k−d+13k−2d−2k−d+13k−2d+1∑j=02d−2khj∗−∑j=2k+1−ddhj∗.

Hibi [12] showed that the inequality

∑j=0m+1hj∗≥∑j=d−mdhj∗ (12)

holds for m=0,…,⌊d2⌋−1. Since 2d−2k−1≤⌊d2⌋−1 for 3d4+1≤k, we can use (12) to finally obtain

fk−1∗−fk∗>0.

□

Proof of Theorem 5

If d = 1 or 2, there is nothing to prove. If 3 ≤ d ≤ 6, then by Theorem 3, either

f0∗≤⋯≤fd2∗≥f3d4∗≥⋯≥fd∗

f0∗≤⋯≤fd2∗≤f3d4∗≥⋯≥fd∗.

For 7 ≤ d ≤ 13, we will show that if fi∗≥fi+1∗, then fi+1∗≥fi+2∗ for d2≤i≤3d4−2. By Theorem 3, this will imply the unimodality of (f0∗,f1∗,…,fd∗). Below, we examine each value of d separately, and make use of the inequality

∑j=1m+1(hj∗−hd+1−j∗)>0, (13)

which holds for m=0,…,⌊d2⌋−1 by [16, Remark 1.2], as well as the nonnegativity of h^*-vectors to deduce that fi+1∗−fi+2∗≥c(fi∗−fi+1∗) for some nonnegative real c in each case.

Suppose that d = 7 and f3∗≥f4∗. Then, by (4) (and h0∗ = 1), we compute

2f4∗−f3∗−f5∗=14h0∗+14h1∗+10h2∗+5h3∗+h4∗−h5∗−h6∗>∑j=13(hj∗−h8−j∗),

hence f4∗−f5∗≥f3∗−f4∗. Likewise, for d = 8 we have

2f5∗−f4∗−f6∗=6h0∗+14h1∗+14h2∗+10h3∗+5h4∗+h5∗−h6∗−h7∗>∑j=13hj∗−h9−j∗,

and similarly for d = 9,

f5∗−f6∗−2(f4∗−f5∗)>∑j=14hj∗−h10−j∗,

and for d = 10,

2f6∗−f5∗−f7∗>2∑j=15hj∗−h11−j∗.

For d = 11, we need to consider two values: i = 5 and i = 6. The claim follows since

f6∗−f7∗−2(f5∗−f6∗)>2∑j=15hj∗−h12−j∗,

and

f7∗−f8∗−45(f6∗−f7∗)>3∑j=15hj∗−h12−j∗.

For d = 12, i also takes two values: i = 6 and i = 7. Now,

f7∗−f8∗−54(f6∗−f7∗)>3∑j=16hj∗−h13−j∗,

and also

f8∗−f9∗−12(f7∗−f8∗)>3∑j=16hj∗−h13−j∗.

Finally, for d = 13, the desired inequality holds for both values i = 6 and i = 7 as weak inequality:

f7∗−f8∗−73(f6∗−f7∗)≥3∑j=16hj∗−h14−j∗

and

2f8∗−f7∗−f9∗≥4∑j=16hj∗−h14−j∗.

To construct a polytope with nonunimodal f^*-vector, we employ a family of simplices introduced by Higashitani [13]. Concretely, denote the jth unit vector by e_j and let

Δw:=conv{0,e1,e2,…,e14,w}

where

w:=(1,1,…,1⏟7,131,131,…,131⏟7,132).

It has h^*-vector

(1,0,0,…,0⏟7,131,0,0,…,0⏟7)

and, via (4), f^*-vector

(16,120,560,1820,4368,8008,11440,13001,12488,11676,11704,10990,7896,3788,1064,132).

□

We record the following consequence of Theorem 5, which follows by the nonnegativity of h^*-vectors.

Corollary 11

Every lattice polytope of degree at most 5 has unimodal f^*-vector.

Proof

Let P be a d-dimensional lattice polytope of degree at most 5. We know from Theorem 5 that f^* is unimodal when d ≤ 13.

Suppose that d ≥ 14. The proof is similar to the proof of Proposition 9, but we need to be a bit more precise with bounds. By Theorems 3(a) and 7, it suffices to show that f⌊d2⌋+i∗≥f⌊d2⌋+i+1∗ implies f⌊d2⌋+i+1∗≥f⌊d2⌋+i+2∗, for i=0,…,⌈d+52⌉−⌊d2⌋−3. Note that ⌈d+52⌉=⌊d2⌋+⌈52⌉, hence i = 0. Arguing as in the proof of Proposition 9, we can reduce the proof to showing that the expression in (5) in Proposition 9 is nonnegative for 0 ≤ j ≤ 5 and i = 0, i.e., that

d−(2j−5)d2−d2−6+j(5−j)≥0. (14)

The inequality in (14) holds for 0 ≤ j ≤ 5 since

d−(2j−5)d2−d2−6+j(5−j)≥d−11.

□

Proof of Theorem 6

Let s:=d+1−g (and 12d+1+d+1−g2≤k≤d). We first consider the case that s is odd; the case s even will be similar. Since hj∗ = 0 for j > s and hj∗=hs−j∗,

fk−1∗−fk∗=∑j=0sd−j+1k−j−d−j+1k−j+1hj∗=∑j=0⌊s2⌋d−j+1k−j−d−j+1k−j+1hj∗+∑j=⌊s2⌋+1sd−j+1k−j−d−j+1k−j+1hj∗=∑j=0⌊s2⌋d−j+1k−j−d−j+1k−j+1+d−s+j+1k−s+j−d−s+j+1k−s+j+1hj∗.

Because we assume k≥12(d+1+⌊s2⌋),

d−j+1k−j−d−j+1k−j+1>0

for 0≤j≤⌊s2⌋. The inequality

d−j+1k−j−d−j+1k−j+1+d−s+j+1k−s+j−d−s+j+1k−s+j+1>0

follows directly if d−s+j+1k−s+j−d−s+j+1k−s+j+1≥0 or k − s + j + 1 < 0. Otherwise, Lemma 10 implies that, for the same range of j,

d−j+1k−j−d−j+1k−j+1+d−s+j+1k−s+j−d−s+j+1k−s+j+1≥0.

In fact, the last inequality is strict for k≥12(d+1+⌊s2⌋); the proof is analogous to that of Lemma 10. Finally we use that hj∗ ≥ 0 (for 0 ≤ j ≤ ⌊s2⌋) and h0∗ = 1 to deduce that fk−1∗−fk∗>0.

The computations in the case s even is very similar. Now we write

fk−1∗−fk∗=∑j=0sd−j+1k−j−d−j+1k−j+1hj∗=∑j=0s2−1d−j+1k−j−d−j+1k−j+1+d−s+j+1k−s+j−d−s+j+1k−s+j+1hj∗+d−s2+1k−s2−d−s2+1k−s2+1hs2∗

and use the same argumentation as in the case s odd. □

3 Concluding remarks

There are many avenues to explore f^*-vectors, e.g., along analogous studies of h^*-vectors, and we hope the above results form an enticing starting point. The techniques in our proof of Theorem 5 do not offer much insight in the case of 14-dimensional lattice polytopes as there are candidate f^*-vectors with corresponding h^*-vectors that satisfy all inequalities discussed in [16]. It is unknown, though, if such polytopes exist.

Higashitani [13, Theorem 1.1] provided examples of d-dimensional polytopes with nonunimodal h^*-vector for all d ≥ 3. Therefore, by Theorem 5 we have examples of polytopes that have such an h^*-vector but their f^*-vector is unimodal. It would be interesting to know if the opposite can be true, that is, if there exist polytopes with unimodal h^*-vector and nonunimodal f^*-vector. By Corollary 11, such polytopes would need to have degree at least 6.

Whenever one detects that a given polynomial is unimodal, it is natural to ask about the stronger property that the polynomial is log concave or, even stronger, real rooted. Our methods do not yield these properties but it would be interesting if one could extend, e.g., Corollary 8 or Proposition 9 along these lines.

Finally, starting with Stapledon’s work [16], there has been much recent attention to symmetric decompositions of h- and h^*-polynomials; see, e.g., [1; 2] and, in particular, [8] where analogous decompositions for f-vectors are discussed. We believe this line of research is worthy of attention with regards to understanding f^*-vectors and the inequalities that hold among their coefficients.

Funding statement: Danai Deligeorgaki was partially supported by the Knut and Alice Wallenberg Foundation.

Acknowledgements

We thank the organizers of Research Encounters in Algebraic and Combinatorial Topics (REACT 2021), where our collaboration got initiated. We are grateful to Luis Ferroni, Katharina Jochemko, Michael Joswig, Matthias Schymura, Liam Solus and Lorenzo Venturello for helpful conversations. We would also like to thank the anonymous referees for the careful reading and suggestions.

Communicated by: M. Joswig

References

[1] C. A. Athanasiadis, Triangulations of simplicial complexes and theta polynomials. Preprint 2022, MR4633746 Zbl 1521.52005Search in Google Scholar

[2] E. Bajo, M. Beck, Boundary h.-polynomials of rational polytopes. SIAM J. Discrete Math. 37 (2023), 1952.1969. MR4633746 Zbl 1521.5200510.1137/22M1508911Search in Google Scholar

[3] M. Beck, S. Robins, Computing the continuous discretely. Springer 2015. MR3410115 Zbl 1339.5200210.1007/978-1-4939-2969-6Search in Google Scholar

[4] L. J. Billera, C. W. Lee, A proof of the sufficiency of McMullen’s conditions for f-vectors of simplicial convex polytopes. J. Combin. Theory Ser. A 31 (1981), 237–255. MR635368 Zbl 0479.5200610.1016/0097-3165(81)90058-3Search in Google Scholar

[5] A. Björner, The unimodality conjecture for convex polytopes. Bull. Amer. Math. Soc. (N.S.) 4 (1981), 187–188. MR598684 Zbl 0458.5200410.1090/S0273-0979-1981-14877-1Search in Google Scholar

[6] A. Björner, Face numbers of complexes and polytopes. In: Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), 1408–1418, Amer. Math. Soc. 1987. MR934345 Zbl 0657.00005Search in Google Scholar

[7] A. Björner, Partial unimodality for f-vectors of simplicial polytopes and spheres. In: Jerusalem combinatorics '93, volume 178 of Contemp. Math., 45–54, Amer. Math. Soc. 1994. MR1310573 Zbl 0815.5201110.1090/conm/178/01891Search in Google Scholar

[8] P. Brändén, L. Solus, Symmetric decompositions and real-rootedness. Int. Math. Res. Not. IMRN (2021), no. 10, 7764–7798. MR4259159 Zbl 1473.0532910.1093/imrn/rnz059Search in Google Scholar

[9] F. Breuer, Ehrhart f^*-coefficients of polytopal complexes are non-negative integers. Electron. J. Combin. 19 (2012), Paper 16, 22 pages. MR3001653 Zbl 1270.5202010.37236/2106Search in Google Scholar

[10] J. Eckhoff, Combinatorial properties of f-vectors of convex polytopes. Normat 54 (2006), 146–159. MR2288936Search in Google Scholar

[11] E. Ehrhart, Sur les polyédres rationnels homothétiques à n dimensions. C. R. Acad. Sci. Paris 254 (1962), 616–618. MR130860 Zbl 0100.27601Search in Google Scholar

[12] T. Hibi, Algebraic combinatorics on convex polytopes. Carslaw Publications, Glebe 1992. MR3183743 Zbl 0772.52008Search in Google Scholar

[13] A. Higashitani, Counterexamples of the conjecture on roots of Ehrhart polynomials. Discrete Comput. Geom. 47 (2012), 618–623. MR2891252 Zbl 1239.5201110.1007/s00454-011-9390-4Search in Google Scholar

[14] J. Hinman, A positive answer to Bárány’s question on face numbers of polytopes. Combinatorica 43 (2023), 953–962. MR4648588 Zbl 0774590810.1007/s00493-023-00042-7Search in Google Scholar

[15] R. P. Stanley, Magic labelings of graphs, symmetric magic squares, systems of parameters, and Cohen–Macaulay rings. Duke Math. J. 43 (1976), 511–531. MR444514 Zbl 0335.0501010.1215/S0012-7094-76-04342-8Search in Google Scholar

[16] A. Stapledon, Inequalities and Ehrhart δ-vectors. Trans. Amer. Math. Soc. 361 (2009), 5615–5626. MR2515826 Zbl 1181.5202410.1090/S0002-9947-09-04776-XSearch in Google Scholar

[17] A. Stapledon, Additive number theory and inequalities in Ehrhart theory. Int. Math. Res. Not. IMRN (2016), no. 5, 1497–1540. MR3509934 Zbl 1342.5201710.1093/imrn/rnv186Search in Google Scholar

Received: 2022-12-01

Published Online: 2024-04-26

Published in Print: 2024-04-25

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/advgeom-2024-0002

Keywords for this article

Lattice polytope; Ehrhart polynomial; Gorenstein polytope; f*-vector; h*-vector; unimodality

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