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

For n >= 2 and fixed k >= 1, we study when an endomorphism f of F-n, 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 m(k )= 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. J. Algebra Comput. (2022) as well as it strengthens results due to Calugareanu-Lam in J. Algebra Appl. (2016). (c) 2023 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons .org /licenses /by /4 .0/).