Perfect polynomials over ${\mathbb{F}}_4$ with less than five prime factors

Abstract

A perfect polynomial $A\in {\mathbb{F}}_4[x]$ is a monic polynomial that equals the sum of its monic divisors. There are no perfect polynomials $A\in {\mathbb{F}}_4[x]$ with exactly $3$ prime divisors, i.e., of the form $A=P^a{Q}^b{R}^c$ where $P,Q,R\in {\mathbb{F}}_4[x]$ are irreducible and $a,b,c$ are positive integers. We characterize the perfect polynomials $A$ with $4$ prime divisors such that one of them has degree $1$. Assume that $A$ has an arbitrary number of distinct prime divisors, we discuss some simple congruence obstructions that arise and we propose three conjectures.