Timeline for Polynomials with Unique Critical Value
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 9, 2016 at 18:57 | vote | accept | Mose Wintner | ||
| Jul 29, 2015 at 2:02 | comment | added | Mose Wintner | Ah, I see. They do furnish another example, but you're right that this is not the generalization I am looking for. Thanks again! | |
| Jul 29, 2015 at 1:46 | comment | added | Jason Starr | 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. | |
| Jul 29, 2015 at 1:39 | comment | added | Mose Wintner | 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 . | |
| Jul 29, 2015 at 1:06 | comment | added | Jason Starr | 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 $0$. | |
| Jul 29, 2015 at 0:43 | comment | added | Jason Starr | 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? | |
| Jul 29, 2015 at 0:38 | history | answered | Mose Wintner | CC BY-SA 3.0 |