Irreducible polynomials with prescribed sums of coefficients

TL;DR

This paper investigates the existence of irreducible polynomials over finite fields with prescribed sums of certain coefficients, proving new results for binary fields and extending to larger fields.

Contribution

It establishes the existence of irreducible polynomials with specific coefficient sum properties over finite fields, providing new proofs and extending known conjectures.

Findings

For $ ext{q}=2$, almost all prescribed sums are achievable with irreducible polynomials.

Provides a new proof of the Hansen-Mullen irreducibility conjecture for $ ext{q}=2$.

For $ ext{q}>2$, shows that not all sums can be realized, but some are always possible.

Abstract

Let $q$ be a power of a prime, let $\mathbb{F}_q$ be the finite field with $q$ elements and let $n \geq 2$. For a polynomial $h(x) \in \mathbb{F}_q[x]$ of degree $n \in \mathbb{N}$ and a subset $W \subseteq [0,n] := \{0, 1, \ldots, n\}$, we define the sum-of-digits function $$S_W(h) = \sum_{w \in W}[x^{w}] h(x)$$ to be the sum of all the coefficients of $x^w$ in $h(x)$ with $w \in W$. In the case when $q = 2$, we prove, except for a few genuine exceptions, that for any $c \in \mathbb{F}_2$ and any $W \subseteq [0,n]$ there exists an irreducible polynomial $P(x)$ of degree $n$ over $\mathbb{F}_2$ such that $S_{W}(P) = c$. In particular, restricting ourselves to the case when $\# W = 1$, we obtain a new proof of the Hansen-Mullen irreducibility conjecture (now a theorem) in the case when $q = 2$. In the case of $q> 2$, we prove that, for any $c \in \mathbb{F}_q$, any $n\geq 2$ and any $W \subseteq [0,n]$, there exists an irreducible polynomial $P(x)$ of degree $n$ such that $S_{W}(P) \neq c$.