Fourier growth of degree $2$ polynomials

TL;DR

This paper establishes bounds on the Fourier coefficients of degree 1 and 2 polynomials over finite fields, specifically analyzing the sum of Fourier coefficients at each level for functions of the form (-1)^{p(x)}.

Contribution

It provides new bounds on the absolute sum of Fourier coefficients at each level for degree 1 and 2 polynomials, advancing understanding of their spectral properties.

Findings

Bounds for Fourier coefficients of degree 1 polynomials established.

Bounds for Fourier coefficients of degree 2 polynomials established.

Results contribute to the spectral analysis of polynomial functions over finite fields.

Abstract

We prove bounds for the absolute sum of all level-$k$ Fourier coefficients for $(-1)^{p(x)}$, where polynomial $p:\mathbf{F}_2^n \to \mathbf{F}_2$ is of degree $1$ or degree $2$.