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Images of multilinear polynomials on n × n upper triangular matrices over infinite fields

2022; Hebrew University of Jerusalem; Volume: 252; Issue: 1 Linguagem: Inglês

10.1007/s11856-022-2350-2

1565-8511

Ivan Gonzales Gargate, Thiago Castilho de Mello,

Advanced Differential Equations and Dynamical Systems

In this paper we prove that the image of multilinear polynomials evaluated on the algebra UTn(K) of n × n upper triangular matrices over an infinite field K equals Jr, a power of its Jacobson ideal J = J(UTn(K)). In particular, this shows that the analogue of the Lvov—Kaplansky conjecture for UTn(K) is true, solving a conjecture of Fagundes and de Mello. To prove that fact, we introduce the notion of commutator-degree of a polynomial and characterize the multilinear polynomials of commutator-degree r in terms of its coefficients. It turns out that the image of a multilinear polynomial f on UTn(K) is Jr if and only if f has commutator-degree r.