Integer points in dilates of polytopes

2025-10-18

AI Summary

This work addresses the long-standing open problem of determining the exact asymptotic upper bound—specifically, the optimal dependence on the degree $d$—for the number of monomials in multivariate polynomial factorization. The core challenge reduces to analyzing the growth rate of integer-point counts in Hadamard polytope dilations. For this class of high-dimensional convex polytopes endowed with sparse algebraic structure, we establish, for the first time, an almost-tight lower bound on the number of integer points in their dilates. Our approach integrates tools from convex geometry, lattice-point enumeration, and combinatorial construction to characterize the asymptotic distribution of integer points in discrete geometry. As a consequence, we prove that the $d$-dependence in the monomial count upper bound is fundamentally optimal, thereby fully resolving the $d$-dependency question posed by Bhargava et al. Moreover, our result yields a significant improvement over prior complexity estimates for polynomial factorization.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Responsible Web: Measurement, analysis, and circumvention of Web censorship

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.

Problem

• Studying growth of integer points in dilated polytopes
• Establishing tight bounds on monomial counts in polynomial factors
• Analyzing integer point dependence on dimension in Hadamard polytopes

Innovation

• Studied integer points in dilated Hadamard polytopes
• Proved new tight bounds on integer point growth
• Established lower bounds via polytope dilation techniques