  Let $f=f(x_1,\dots,x_m)$ be a multilinear polynomial over a field $F$. An
$F$-algebra $A$ is said to be $f$-zpd ($f$-zero product determined) if every
$m$-linear functional $\varphi\colon A^{m}\rightarrow F$ which preserves zeros
of $f$ is of the form $\varphi(a_1,\dots,a_m)=\tau(f(a_1,\dots,a_m))$ for some
linear functional $\tau$ on $A$. We are primarily interested in the question
whether the matrix algebra $M_d(F)$ is $f$-zpd. While the answer is negative in
general, we provide several families of polynomials for which it is positive.
We also consider a related problem on the form of a multilinear polynomial
$g=g(x_1,\dots,x_m)$ with the property that every zero of $f$ in $M_d(F)^{m}$
is a zero of $g$. Under the assumption that $m<2d-3$, we show that $g$ and $f$
are linearly dependent.
