Roots of sparse polynomials over a finite field

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$2(q-1)^{1-\unicode[STIX]{x1D700}}C^{\unicode[STIX]{x1D700}}$, where$\unicode[STIX]{x1D700}=1/(t-1)$and$C$is the size of the largest coset in$\mathbb{F}_{q}^{\ast }$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 Canettiet 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$\mathbb{F}_{p}^{\ast }$when$C=1$.