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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 $Ω\left(\frac{\log p}{\log \log p}\right)$ distinct roots in $\mathbb{F}_p$.