Integer Points in Dilates of Polytopes

Abstract

In this paper we study how the number of integer points in a polytope grows as we dilate the polytope. We prove new and essentially tight bounds on this quantity by specifically studying dilates of the Hadamard polytope. 
The motivation for studying this question comes from the question of understanding the maximal number of monomials in a factor of a multivariate polynomial of s monomials. A recent result by Bhargava, Saraf and Volkovich [Bhargava et al., 2020] showed that if f is an n-variate polynomial, where each variable has degree d, and f has s monomials, then any factor of f has at most s^{O(d² log n)} monomials. The key technical ingredient of their proof was to show that any polytope with s vertices, where each vertex lies in {0,..,d}ⁿ can have at most s^{O(d² log n)} integer points. The precise dependence on d of the number of integer points was left open. We show that this bound, and in particular the dependence on d is essentially tight by studying dilates of the Hadamard polytope and proving new lower bounds on its number of integer points.