Maps preserving zeros of a polynomial

Let $\text{A}$ be an algebra and let $f(x_1,\mathrm{.}\mathrm{.}\mathrm{.},x_d)$ be a multilinear polynomial in noncommuting indeterminates $x_i$. We consider the problem of describing linear maps $\varphi :\text{A}\to \text{A}$ that preserve zeros of $f$. Under certain technical restrictions we solve the problem for general polynomials $f$ in the case where $\text{A}=M_n(F)$. We also consider quite general algebras $\text{A}$, but only for specific polynomials $f$.