Sparse Univariate Polynomials with Many Roots Over Finite Fields

TL;DR

This paper constructs explicit sparse univariate polynomials over finite fields with a number of roots approaching theoretical bounds, providing new insights into root structures and their complexity over different finite fields.

Contribution

It presents explicit constructions of sparse polynomials with many roots over finite fields, approaching known upper bounds, and explores their properties over prime and extension fields.

Findings

Explicit polynomials with roots approaching theoretical bounds over extension fields.

Computational evidence suggests difficulty in constructing such polynomials over prime fields.

Under GRH, explicit trinomials with many roots in prime fields are demonstrated.

Abstract

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound of $2(q-1)^{\frac{t-2}{t-1}}$ on the number of cosets in $\mathbb{F}^*_q$ needed to cover the roots of $f$ in $\mathbb{F}^*_q$. Here, we give explicit $f$ with root structure approaching this bound: For $q$ a $(t-1)$-st power of a prime we give an explicit $t$-nomial vanishing on $q^{\frac{t-2}{t-1}}$ distinct cosets of $\mathbb{F}^*_q$. Over prime fields $\mathbb{F}_p$, computational data we provide suggests that it is harder to construct explicit sparse polynomials with many roots. Nevertheless, assuming the Generalized Riemann Hypothesis, we find explicit trinomials having $\Omega\left(\frac{\log p}{\log \log p}\right)$ distinct roots in $\mathbb{F}_p$.