Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence

Abstract

For n ≥ 2 and fixed k ≥ 1, we study when an endomorphism f of Fn, where F is an arbitrary field, can be decomposed as t + m where t is a root of the unity endomorphism and m is a nilpotent endomorphism with mk = 0. For fields of prime characteristic, we show that this decomposition holds as soon as the characteristic polynomial of f is algebraic over its base field and the rank of f is at least n k , and we present several examples that show that the decomposition does not hold in general. Furthermore, we completely solve this decomposition problem for k = 2 and nilpotent endomorphisms over arbitrary fields (even over division rings). This somewhat continues our recent publications in Linear Multilinear Algebra (2022) and Int.