Irreducible Polynomials with Coefficients in an Affine Algebraic Set

TL;DR

This paper establishes bounds on the count of monic irreducible polynomials over finite fields with coefficients constrained within a specific affine algebraic set, advancing understanding of polynomial distributions in algebraic geometry.

Contribution

It provides explicit error bounds for the number of such irreducible polynomials with coefficients in a given affine algebraic set over finite fields.

Findings

Derived bounds for polynomial counts in affine algebraic sets

Quantified the distribution of irreducible polynomials with constrained coefficients

Extended algebraic geometry methods to polynomial enumeration

Abstract

In this paper, we give error bounds on the number of monic irreducible polynomials $a_0+a_1x+\dots+a_{n-1}x^{n-1}+x^n$ over a finite field $\mathbb{F}_q$ of degree $n$ with $(a_0, a_1, \dots, a_{n-1}, 1)$ lying in a fixed affine algebraic set $V$ of points in $\mathbb{F}_q^{n+1}$.