Let f = f(X1, …, Xm) be a non-commutative polynomial with coefficients in a field F. We will discuss various questions concerning the image of f in an F-algebra A, which is defined to be the set f(A) = {f(a1, …, am) | a1, …, am ∈ A}. A special emphasis will be on the Waring-type problem, asking about the existence of a positive integer N (independent of f, provided that f is neither an identity nor a central polynomial of A) such that every element, or at least every commutator, in A is a linear combination of N elements from f(A). We are primarily interested in the case where A = Mn(F), but some other algebras will also be considered.
This video was produced by the Universidade de São Paulo, as part of the LieJor Online Seminar: Algebras, Representations, and Applications.
