Fixed points of the sum of divisors function on $F_2[x]$

Abstract

We work an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points $F$ of the sum of divisors function $\sigma :F_2[x]\mapsto F_2[x]$ (defined \emph{mutatis mutandi} like the usual sum of divisors over the integers) of the form $F:=A^2\cdot S$, $S$ square-free, with $\omega (S)\le 3$, coprime with $A$, for $A$ even, of whatever degree, under some conditions. This gives a characterization of $5$ of the $11$ known fixed points of $\sigma$ in $F_2[x]$