Maps preserving zeros of a polynomial

Let A be an algebra and let f(x1,…,xd) 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.