Evaluations of Noncommutative Polynomials on Algebras: Methods and Problems, and the L'vov-Kaplansky Conjecture

Abstract

Let 𝑝 be a polynomial in several non-commuting variables with coefficients in a field 𝛫 of arbitrary characteristic. It has been conjectured that for any 𝑛, for 𝑝 multilinear, the image of 𝑝 evaluated on the set Mₙ(𝛫) of 𝑛 by 𝑛 matrices is either zero, or the set of scalar matrices, or the set slₙ(𝛫) of matrices of trace 0, or all of Mₙ(𝛫). This expository paper describes research on this problem and related areas. We discuss the solution of this conjecture for 𝑛 = 2 in Section 2, some decisive results for 𝑛 = 3 in Section 3, and partial information for 𝑛 ≥ 3 in Section 4, also for non-multilinear polynomials. In addition, we consider the case of 𝛫 not algebraically closed, and polynomials evaluated on other finite-dimensional simple algebras (in particular, the algebra of the quaternions). This review recollects results and technical material of our previous papers, as well as new results of other research, and applies them in a new context. This article also explains the role of the Deligne trick, which is related to some nonassociative cases in new situations, underlying our earlier, more straightforward approach. We pose some problems for future generalizations and point out possible generalizations in the present state of the art, and on the other hand, provide counterexamples showing the boundaries of generalizations.