Roots of Sparse Polynomials over a Finite Field

TL;DR

This paper establishes new bounds on the number of roots of sparse polynomials over finite fields, improving previous results and proposing a conjecture for prime fields.

Contribution

It introduces a novel upper bound on the roots of t-nomials over finite fields based on coset structures and provides a computable parameter for this bound.

Findings

Bound on roots: at most 2(q-1)^{1-ε} C^ε roots for t-nomials.

Introduction of a number-theoretic parameter for bounding C.

Conjecture: t-nomials over prime fields have O(t log p) roots when C=1.

Abstract

For a $t$-nomial $f(x) = \sum_{i = 1}^t c_i x^{a_i} \in \mathbb{F}_q[x]$, we show that the number of distinct, nonzero roots of $f$ is bounded above by $2 (q-1)^{1-\varepsilon} C^\varepsilon$, where $\varepsilon = 1/(t-1)$ and $C$ is the size of the largest coset in $\mathbb{F}_q^*$ on which $f$ vanishes completely. Additionally, we describe a number-theoretic parameter depending only on $q$ and the exponents $a_i$ which provides a general and easily-computable upper bound for $C$. We thus obtain a strict improvement over an earlier bound of Canetti et al.\ which is related to the uniformity of the Diffie-Hellman distribution. Finally, we conjecture that $t$-nomials over prime fields have only $O(t \log p)$ roots in $\mathbb{F}_p^*$ when $C = 1$.