Fibre Products of Supersingular Curves and the Enumeration of Irreducible Polynomials with Prescribed Coefficients

TL;DR

This paper derives formulas for counting irreducible polynomials over finite fields with specific zero coefficients, using algebraic curves and Fourier analysis, revealing periodicity in the formulas.

Contribution

It introduces a novel approach combining fibre products of supersingular curves and Fourier analysis to count irreducible polynomials with prescribed coefficients.

Findings

Formulas for irreducible polynomials with certain zero coefficients over finite fields.

Connection between algebraic curves and polynomial enumeration.

Periodicity of 24 in the counting formulas.

Abstract

For any positive integers $n\geq 3, r\geq 1$ we present formulae for the number of irreducible polynomials of degree $n$ over the finite field $\mathbb{F}_{2^r}$ where the coefficients of $x^{n-1}$, $x^{n-2}$ and $x^{n-3}$ are zero. Our proofs involve counting the number of points on certain algebraic curves over finite fields, a technique which arose from Fourier-analysing the known formulae for the $\mathbb{F}_2$ base field cases, reverse-engineering an economical new proof and then extending it. This approach gives rise to fibre products of supersingular curves and makes explicit why the formulae have period $24$ in $n$.