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

In this paper we prove that the image of multilinear polynomials evaluated on the algebra UT n ( K ) of n × n upper triangular matrices over an infinite field K equals J r , a power of its Jacobson ideal J = J ( UT n ( K )). In particular, this shows that the analogue of the Lvov—Kaplansky conjecture for UT n ( 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 UT n ( K ) is J r if and only if f has commutator-degree r .