Sparse Univariate Polynomials with Many Roots Over Finite Fields

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 $\mathbb{F}p$.