My question is extremely simple to state: I am looking for a characterization of multivariate complex polynomials $f$ such that $f(Sing(f))=\{0\}$. My motivation is that I recently read somewhere that any polynomial only possessing one degenerate fiber defines an isotrivial family away from that fiber. If that question ends up being easy, is there a good characterization known of multivariate polynomials over an arbitrary algebraically closed field only possessing one critical value?