Let f=f(x1,…,xm) 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 φ:Am→F which preserves zeros
of f is of the form φ(a1,…,am)=τ(f(a1,…,am)) for some
linear functional τ on A. We are primarily interested in the question
whether the matrix algebra Md(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(x1,…,xm) with the property that every zero of f in Md(F)m
is a zero of g. Under the assumption that m<2d−3, we show that g and f
are linearly dependent