The Number of Irreducible Polynomials with the First Two Coefficients Fixed over Finite Fields of Odd Characteristic

TL;DR

This paper provides a new proof for counting irreducible polynomials with fixed initial coefficients over finite fields of odd characteristic, using algebraic geometry techniques involving point counts on curves.

Contribution

It introduces a novel proof method relating polynomial enumeration to point counting on algebraic curves, expanding the toolkit for finite field polynomial analysis.

Findings

Derived a new proof for Kuz'min's result

Connected polynomial counting to algebraic curve point counts

Calculated the L-polynomial of the associated curve

Abstract

In this paper we give a different proof of Kuz'min's result on the number of irreducible polynomials with the first two coefficients fixed. Our technique is to relate the question to the number of points on a curve, and to calculate the L-polynomial of the curve.