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?
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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?
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1$\begingroup$ That is false in positive characteristic, cf. my answer to the following question: When is the kernel of the etale fundamental group in a fibration abelian? $\endgroup$ Commented Jul 29, 2015 at 0:43
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1$\begingroup$ Let me just spell that out, because I just confused myself about how this works! Let $k$ be algebraically closed of characteristic $2$. Let $a$, $b$ be elements in $k$ such that none of $a$, $b$, nor $a+b$ is $0$. Then the correct dehomogenized polynomial on $\mathbb{A}^2_k$ is $f(x,y) = xy(1-ax-by)(1+bx+ay)/(a+b)^2$. This is not weighted homogeneous (unless you allow both $x$ and $y$ to have weights $0$, in which case every polynomial is weighted homogeneous). Yet the critical locus is precisely $0$. $\endgroup$ Commented Jul 29, 2015 at 1:06
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1$\begingroup$ Thanks Jason! I figured the result was too special to generalize nicely to positive characteristic. I see now that I really should have thought and googled more thoroughly before posting this question on MathOverflow, since I just found this paper which claims to partially, weakly generalize Saito's aforementioned result to charateristic $p$: mathematik.uni-kl.de/~pfister/Artikel/AouiraPfister-MZ91.pdf . $\endgroup$ Commented Jul 29, 2015 at 1:39
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$\begingroup$ Probably you already noticed this, but the article of Aouira and Pfister does not include my example above. They are generalizing Saito's criterion in a different direction: trying to characterize when an isolated singularity has a non-nilpotent vector field, rather than trying to characterize when the critical locus is a single point. $\endgroup$ Commented Jul 29, 2015 at 1:46
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$\begingroup$ Ah, I see. They do furnish another example, but you're right that this is not the generalization I am looking for. Thanks again! $\endgroup$ Commented Jul 29, 2015 at 2:02