Integer points in dilates of polytopes
Shubhangi Saraf, Narmada Varadarajan
TL;DR
This paper investigates how the number of integer points in a polytope grows under dilation, focusing on the Hadamard polytope to test tightness of known bounds. It introduces an algebraic characterization of lattice points in $P_{\mathrm{Had}}$, showing that the support of any integer point is a subspace of $\mathbb F_2^m$ and that $T(v)=\mathrm{supp}(v)^{\perp}+b$, implying $|P_{\mathrm{Had}}\cap \mathbb Z^n|=n^{\Theta(\log n)}$. For dilates, the authors derive lower bounds on $|dP_{\mathrm{Had}}\cap \mathbb Z^n|$ across ranges of $d< n \log n$, demonstrating growth from $n^{\Omega(d \log n)}$ in the small-d regime to exponential-in-$n$ bounds in larger regimes, including $(d^{2\varepsilon}/4)^n$ and $n^{\Omega(\log^2 n)}$. These results show that both the $d^2$ and $\log n$ factors in the exponent of the previously known upper bound are essentially tight, and they connect discrete polytope counting with sparse polynomial factorization in algebraic complexity.
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. Our motivation for studying this quantity comes from the problem of understanding the maximal number of monomials in a factor of a multivariate polynomial with $s$ monomials. A recent result by Bhargava, Saraf, and Volkovich 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 $s^{O(d^2 \log n)}$ 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 $s^{O(d^2 \log n)}$ integer points. The precise dependence on $d$ of the number of integer points was left open. We show that this bound, particularly the dependence on $d$, is essentially tight by studying dilates of the Hadamard polytope and proving new lower bounds on the number of its integer points.