Quasi-random multilinear polynomials

TL;DR

This paper investigates bounds on the maximum absolute value of multilinear Littlewood polynomials with specified monomials, connecting the problem to quasi-random structures, statistical mechanics, and advanced probabilistic inequalities.

Contribution

It provides new upper and lower bounds for these polynomials' maxima, especially for low-degree cases, and links the problem to various mathematical theories.

Findings

Bounds are tight for degree below log n

Connections established to quasi-random graphs and hypergraphs

Methods include Gale-Berlekamp game and Khintchine inequalities

Abstract

We consider multilinear Littlewood polynomials, polynomials in $n$ variables in which a specified set of monomials $U$ have $\pm 1$ coefficients, and all other coefficients are $0$. We provide upper and lower bounds (which are close for $U$ of degree below $\log n$) on the minimum, over polynomials $h$ consistent with $U$, of the maximum of $|h|$ over $\pm 1$ assignments to the variables. (This is a variant of a question posed by Erd\"os regarding the maximum on the unit disk of univariate polynomials of given degree with unit coefficients.) We outline connections to the theory of quasi-random graphs and hypergraphs, and to statistical mechanics models. Our methods rely on the analysis of the Gale-Berlekamp game; on the constructive side of the generic chaining method; on a Khintchine-type inequality for polynomials of degree greater than $1$; and on Bernstein's approximation theory inequality.