Lattice polytopes of degree 2

TL;DR

This paper extends Scott's volume bound for lattice polygons with interior points to higher-dimensional lattice polytopes of degree 2, showing finiteness results for associated quadratic polynomials.

Contribution

It generalizes Scott's theorem to higher dimensions and establishes finiteness of certain quadratic polynomials related to lattice polytopes.

Findings

Normalized volume bounds for degree 2 lattice polytopes in higher dimensions

Finiteness of quadratic polynomials as h*-polynomials of lattice polytopes

Extension of known polygon results to polytopes in higher dimensions

Abstract

A theorem of Scott gives an upper bound for the normalized volume of lattice polygons with exactly $i>0$ interior lattice points. We will show that the same bound is true for the normalized volume of lattice polytopes of degree 2 even in higher dimensions. In particular, there is only a finite number of quadratic polynomials with fixed leading coefficient being the $h^*$-polynomial of a lattice polytope.