Fixed points of the sum of divisors function on \({{\mathbb{F}}}_2[x]\)

We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points \(F\) of the sum of divisors function \(\sigma : {\mathbb{F}}_2[x] \mapsto {\mathbb{F}}_2[x]\) (defined mutatis mutandi like the usual sum of divisors over the integers) of the form \(F := A^2 \cdot S\), \(S\) square-free, with \(\omega(S) \leq 3\), 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 \({\mathbb{F}}_2[x]\).