f-zpd algebras and a multilinear Nullstellensatz

Abstract: 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.