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

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.

Incorporated referee comments. Accepted for publication in Finite Fields Appl