Smooth polynomials with several prescribed coefficients

TL;DR

This paper studies the distribution of m-smooth polynomials over finite fields with specific coefficients, using advanced character sum techniques and Bourgains's argument.

Contribution

It introduces new methods combining character sum estimates and Bourgains's argument to analyze prescribed coefficients in smooth polynomials.

Findings

Distribution patterns of m-smooth polynomials with prescribed coefficients

Application of double character sums to polynomial smoothness

Extension of Bourgains's argument to polynomial settings

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, Bourgains's argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials.