Detecting the integer decomposition property and Ehrhart unimodality in reflexive simplices

Abstract

A long-standing open conjecture in combinatorics asserts that a Gorenstein lattice polytope with the integer decomposition property (IDP) has a unimodal (Ehrhart) h * -polynomial. This conjecture can be viewed as a strengthening of a previously disproved conjecture which stated that any Gorenstein lattice polytope has a unimodal h * -polynomial. The first counterexamples to unimodality for Gorenstein lattice polytopes were given in even dimensions greater than five by Mustaţǎ and Payne, and this was extended to all dimensions greater than five by Payne. While there exist numerous examples in support of the conjecture that IDP reflexives are h * -unimodal, its validity has not yet been considered for families of reflexive lattice simplices that closely generalize Payne's counterexamples. The main purpose of this work is to prove that the former conjecture does indeed hold for a natural generalization of Payne's examples. The second purpose of this work is to extend this investigation to a broader class of lattice simplices, for which we present new results and open problems.

Simplices for numeral systems

“…then we see by the decomposed presentation of f (4,3) in the third equality that f 3,0 (4,3) = 1 + 10z + 10z 2 + z 3 , f 3,1 (4,3) = 3 + 12z + 6z 2 , and f 3,2 (4,3) = 6 + 12z + 3z 2 . Then, by Theorem 4.2, we know that h * (B (4,3) ; z) = (1 + 10z + 10z 2 + z 3 ) + z(9 + 24z + 9z 2 ),…”

“…Moreover, it turns out the h * -polynomial of this n-simplex can be computed in a recursive fashion in terms of the numeral representations of the first a n nonnegative integers. This recursive formula is presented in Proposition 3.2, but it is essentially due to the following theorem of [4].…”

“…We can recompute this polynomial using the formula proved in Theorem 4.2. If we expand the polynomial f (4,3) and write it diagrammatically as f (4,3) = (1 + z + z 2 + z 3 ) 3 , = 1 + 3z + 6z 2 + 10z 3 + 12z 4 + 12z 5 + 10z 6 + 6z 7 + 3z 8 + z 9 ,…”

“…Example 4.1 presents the h * -polynomial of the base-4 3-simplex B (4,3) in terms of the average weighted height statistic. We can recompute this polynomial using the formula proved in Theorem 4.2.…”

“…The distributional (and related algebraic) properties of the h * -polynomials for the simplices ∆ (1,q) were recently studied in terms of the arithmetic structure of q [3,4]. In particular, [4, Theorem 2.5] provides an arithmetic formula for h * (∆ (1,q) ; z) in terms of q which the authors use to prove unimodality of the h * -polynomial in some special cases.…”