Smooth polynomials with several prescribed coefficients

Abstract

Let $\mathbb{F}_q[t]$ be the polynomial ring over the finite field $\mathbb{F}_q$ of $q$ elements. A polynomial in $\mathbb{F}_q[t]$ is called $m$-smooth (or $m$-friable) if all its irreducible factors are of degree at most $m$. In this paper, we investigate the distribution of $m$-smooth (or $m$-friable) polynomials with prescribed coefficients. Our technique is based on character sum estimates on smooth (friable) polynomials proved by the analytic approach of Panario, Gourdon, Flajolet (1998), Bourgains's argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials