Lattice polytopes having h^*-polynomials with given degree and linear coefficient

TL;DR

This paper establishes a combinatorial characterization of lattice polytopes with specific h^*-polynomial properties, showing they are pyramids over lower-dimensional polytopes under certain dimension conditions.

Contribution

It generalizes Batyrev's theorem by providing a new combinatorial proof that relates the dimension of a lattice polytope to its h^*-polynomial coefficients.

Findings

Lattice polytopes with certain h^*-polynomial degrees are pyramids over lower-dimensional polytopes.

The dimension threshold for this pyramid structure depends on the polynomial's degree and linear coefficient.

The proof is purely combinatorial, extending previous results in the field.

Abstract

The h^*-polynomial of a lattice polytope is the numerator of the generating function of the Ehrhart polynomial. Let P be a lattice polytope with h^*-polynomial of degree d and with linear coefficient h^*_1. We show that P has to be a lattice pyramid over a lower-dimensional lattice polytope, if the dimension of P is greater or equal to h^*_1 (2d+1) + 4d-1. This result has a purely combinatorial proof and generalizes a recent theorem of Batyrev.