The Fourier structure of low degree polynomials

TL;DR

This paper investigates the Fourier coefficient structure of low degree multivariate polynomials over finite fields, revealing that key properties are closely related even for higher degrees, extending known quadratic relations.

Contribution

It extends the known relations between Fourier coefficient properties from quadratic to higher degree polynomials over finite fields.

Findings

The three Fourier properties are equivalent up to exponential factors in degree d.

Relations known for quadratic polynomials are generalized to higher degrees.

Provides a deeper understanding of polynomial Fourier spectra over finite fields.

Abstract

We study the structure of the Fourier coefficients of low degree multivariate polynomials over finite fields. We consider three properties: (i) the number of nonzero Fourier coefficients; (ii) the sum of the absolute value of the Fourier coefficients; and (iii) the size of the linear subspace spanned by the nonzero Fourier coefficients. For quadratic polynomials, tight relations are known between all three quantities. In this work, we extend this relation to higher degree polynomials. Specifically, for degree $d$ polynomials, we show that the three quantities are equivalent up to factors exponential in $d$.