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00925njm 22002777a 4500 dc Danchev, Peter author García González, Esther author Gómez-Lozano, Miguel Ángel author 2023-07-10 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. Peter Danchev, Esther García, Miguel Gómez Lozano, Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence, Linear Algebra and its Applications, Volume 676, 2023, Pages 44-55, ISSN 0024-3795, https://doi.org/10.1016/j.laa.2023.07.005 https://hdl.handle.net/10630/27998 10.1016/j.laa.2023.07.005 Anillos (Álgebra) Números complejos Decompositions of endomorphisms into a sum of roots of the unity and nilpotent endomorphisms of fixed nilpotence.