Irreducible polynomials with prescribed sums of coefficients

Abstract

Let $q$ be a power of a prime, let ${\mathbb{F}}_q$ be the finite field with $q$ elements and let $n\ge 2$. For a polynomial $h(x)\in {\mathbb{F}}_q[x]$ of degree $n\in \mathbb{N}$ and a subset $W\subseteq [0,n]:=\{0,1,\dots ,n\}$, 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 $n\ge 2$ and any $W\subseteq [0,n]$, there exists an irreducible polynomial $P(x)$ of degree $n$ such that $S_W(P)\ne c$.

arXiv admin note: text overlap with arXiv:1604.04023

Cited by in corpus (1)

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