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

TL;DR

This paper investigates fixed points of the sum of divisors function over polynomial rings in F_2[x], characterizing some known fixed points and providing new insights into their structure.

Contribution

It offers a partial characterization of fixed points of the sum of divisors function over F_2[x], extending classical arithmetic problems to polynomial rings.

Findings

Characterization of 5 out of 11 known fixed points

Conditions for fixed points of the form A^2 * S with specific properties

Insights into the structure of fixed points in polynomial rings over F_2

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 $\omega(S) \leq 3$, 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 $F_2[x]$