Резюме. Let f(x₁, …, x_m) be a polynomial in noncommutative variables over an infinite field K. If A is a K-algebra, it defines in a natural way a map A^m → A. If the polynomial f is multilinear, the famous Lvov-Kaplansky conjecture asks whether the image of a multilinear polynomial on a matrix algebra is a vector subspace. Solutions to this problem are known only for n = 2 or m = 2 with partial results for m = 3 and n = 3.

In this talk, we survey these results and discuss possible variations for this problem in the associative and nonassociative settings.

In particular, we discuss a joint work with I. Gargate about images of multilinear polynomials on upper triangular matrices.