Irreducible polynomials with several prescribed coefficients

TL;DR

This paper proves that for large finite fields, there exist irreducible polynomials with a significant number of prescribed coefficients, extending previous bounds and using Bourgain's technique.

Contribution

It improves existing bounds on the number of prescribed coefficients in irreducible polynomials over finite fields, applying Bourgain's method for broader cases.

Findings

Existence of irreducible polynomials with up to approximately quarter of the coefficients prescribed for large q.

Extension of bounds from rom rom previous rom Pollack's rom work.

Applicable for any finite field size q with improved coefficient constraints.

Abstract

We study the number of irreducible polynomials over $\mathbf{F}_{q}$ with some coefficients prescribed. Using the technique developed by Bourgain, we show that there is an irreducible polynomial of degree $n$ with $r$ coefficients prescribed in any location when $r \leq \left[\left(1/4 - \epsilon\right)n \right]$ for any $\epsilon>0$ and $q$ is large; and when $r\leq\delta n$ for some $\delta>0$ and for any $q$. The result is improved from the earlier work of Pollack that the similar result holds for $r\leq\left[(1-\epsilon)\sqrt{n}\right]$.