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?



Polynomials with Unique Critical Value
Whoops, figured it out, at least in the case of complex polynomials.


We're looking for the set of $f$ such that $Z(f)=f^{-1}(0)\subseteq Z(j(f))$, where $j(f)$ denotes the Jacobian ideal of $f$. However, a result of Saito shows that this is true if and only if $f$ is weighted homogeneous. Does anyone know of an analog for positive characteristic?


