Critical operators for the degree of the minimal polynomial of derivations restricted to Grassmann spaces
Let V be a finite dimension vector space. For a linear operator on V , f, D(f) denotes the restriction of the derivation associated with f to the mth Grassmann space of V. In [Cyclic Spaces for Grassmann Derivatives and Additive Theory, Bull. London Math. Soc. 26(1994) 140-146] Dias da Silva and Hamidoune obtained a lower bound for the degree of the minimal polynomial of D(f), over an arbitrary field. Over a field of zero characteristic that lower bound is given by deg (PD(f)) ≥ m(deg(Pf)−m)+1. Using additive number theory results, results on the elementary divisors of D(f) and methods presented by Marcus and Ali in [Minimal Polynomials of Additive Commutators and Jordan Products, J. Algebra 22(1972) 12-33] we obtain a characterization of equality cases in the former inequality, over a field of zero characteristic, whenever m does not exceed the number of distinct eigenvalues of f.
Pré-publicações do Departamento de Matemática da Universidade de Coimbra