On Fagundes-Mello conjecture (Q2057239)

The famous Lvov-Kaplansky conjecture states that the image of a multilinear polynomial in non-commutative variables over a field \(K\) on the matrix algebra \(M_n(K)\), \(n\geq 2\), is a vector space. Replacing the algebra \(M_n(K)\) with the upper triangular matrix algebra \(T_n(K)\), Fagundes-de Mello conjecture [\textit{P. S. Fagundes} and \textit{T. C. de Mello}, Oper. Matrices 13, No. 1, 283--292 (2019; Zbl 1432.16023)] states that the image of a multilinear polynomial on the algebra \(T_n(K)\), \(n\geq 2\), is a vector space. In the paper under review the authors confirm the conjecture when the base field \(K\) is infinite or if it is finite but has sufficiently many elements. For the proof the authors define the \(\beta\)-index of a nonzero multilinear polynomial and show that the image of the multilinear polynomial \(p(x_1,\ldots,x_m)\) depends on its index \(\beta(p)\): \par (i) If \(\beta(p)=0\), then \(p(T_n(K))=T_n(K)\); \par (ii) If \(1\leq \beta(p)=t<n\), then \(p(T_n(K))=T_n(K)^{(t-1)}\), where \(T_n(K)^{(t-1)}=J^t(T_n(K))\) and \(J(T_n(K))\) is the Jacobson radical of \(T_n(K)\); \par (iii) If \(\beta(p)\geq n\), then \(p(T_n(K))=0\); \par The Fagundes-de Mello conjecture was solved for infinite fields independently and with other methods in [\textit{I. G. Gargate} and \textit{C. de Mello}, ``Images of multilinear polynomials on \(n\times n\) upper triangular matrices over infinite fields'', Preprint, \url{arXiv: 2106.12726}]. Comparing both proofs one can conclude that the equality \(\beta(p)=t\) is equivalent to the fact that \(p(x_1,\ldots,x_n)\) belongs to \(C^t(K\langle X\rangle)\setminus C^{t+1}(K\langle X\rangle)\), where \(C(K\langle X\rangle)=([x_1,x_2])^T\) is the T-ideal of the free associative algebra \(K\langle X\rangle\) generated by the commutator \([x_1,x_2]\).