Maps preserving zeros of a polynomial

Abstract

Let A be an algebra and let f (x1, . . . , x d ) be a multilinear polynomial in noncommuting indeterminates xi. We consider the problem of describing linear maps φ : A → A that preserve zeros of f . Under certain technical restrictions we solve the problem for general polynomials f in the case where A = Mn(F ). We also consider quite general algebras A, but only for specific polynomials f .