Irreducible polynomials over $\mathbb{F}_{2^r}$ with three prescribed coefficients

TL;DR

This paper investigates the periodic behavior of the count of monic irreducible polynomials over finite fields with three fixed coefficients, revealing characteristic-specific phenomena linked to supersingular curves.

Contribution

It establishes the periodicity of the number of such polynomials over fields with characteristics 2 and 5, connecting algebraic properties to geometric supersingularity.

Findings

Periodicity of 24 over $ ext{F}_{2^r}$

Periodicity of 60 over $ ext{F}_{5^r}$

Unique phenomena in characteristics 2 and 5

Abstract

For any positive integers $n \ge 3$ and $r \ge 1$, we prove that the number of monic irreducible polynomials of degree $n$ over $\mathbb{F}_{2^r}$ in which the coefficients of $T^{n-1}$, $T^{n-2}$ and $T^{n-3}$ are prescribed has period $24$ as a function of $n$, after a suitable normalization. A similar result holds over $\mathbb{F}_{5^r}$, with the period being $60$. We also show that this is a phenomena unique to characteristics $2$ and $5$. The result is strongly related to the supersingularity of certain curves associated with cyclotomic function fields, and in particular it complements an equidistribution result of Katz.