Timeline for Every real variety contains non-singular points
Current License: CC BY-SA 3.0
15 events
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| Mar 4, 2017 at 7:15 | history | edited | Dima Pasechnik | CC BY-SA 3.0 |
removed greetings/signature
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| S Mar 4, 2017 at 6:52 | history | suggested | Martin Sleziak |
removed deprecated (geometry) tag - see the tag info: http://mathoverflow.net/tags/geometry/info
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| Mar 4, 2017 at 6:35 | review | Suggested edits | |||
| S Mar 4, 2017 at 6:52 | |||||
| Feb 22, 2017 at 22:03 | vote | accept | Adam Sheffer | ||
| Feb 20, 2017 at 23:49 | answer | added | Terry Tao | timeline score: 12 | |
| Feb 19, 2017 at 20:03 | comment | added | nfdc23 | I don't meant that your argument is definitely false, but rather than it seems to be making a leap from "dimension $n-1$" to asserting $I(V)$ is principal; I have never studied real algebraic geometry in a serious way, so I have no idea if that principality claim is true, or if true then whether it is hard to prove. The complexification as you now define it is reduced and so generically smooth, with "smooth locus" complementary to the zero locus of an ideal over $\mathbf{R}$ (i.e., comes from a Zariski-dense open over $\mathbf{R}$), there's no easy reason that should have $\mathbf{R}$-points. | |
| Feb 19, 2017 at 19:26 | comment | added | Adam Sheffer | Also, as I said below, it seems that as an outsider I'm not using the standard definitions. I'll be happy to add any additional explanations for what I mean. | |
| Feb 19, 2017 at 19:01 | comment | added | Adam Sheffer | Thank you nfdc23. I see that I need to state my definitions more clearly. By complexification of V, I mean $Z_{\mathbb C}(I(V))$, where by $Z_{\mathbb C}$ I mean the intersection of the zero sets of the polynomials in the ideal, over ${\mathbb C}$. Equivalently, this is the smallest complex variety that contains V. I had in mind a 1957 Annals paper of Whitney that studied this definition. I did read the B-C-R introduction a while back. But I'm still not sure if I made any claims that are false. If I made any, I'd very much appreciate it if you could point what these are. | |
| Feb 19, 2017 at 17:55 | comment | added | nfdc23 | That "dimension $n-1$" implies $I(V)$ is principal is not explained (the analogue for schemes or over an algebraically closed field does not formally imply it). Your comment to JSE about complexification of $V$ (not a standard operation: one can make definitions, but they have real subtleties) and "real part" of an ideal suggest that the delicate nature of the passage between algebra and geometry in real algebraic geometry is something you may not be fully aware of; see the Intro of B-C-R. Also, problems in real algebraic geometry are almost never solved by importing results over $\mathbf{C}$. | |
| Feb 19, 2017 at 14:24 | answer | added | Sean Lawton | timeline score: 2 | |
| Feb 19, 2017 at 7:40 | comment | added | Adam Sheffer | Thanks for the warning Jordan! In this specific case, can't I simply take the complexification of V, claim this over the complex, and then taking the real part of the ideal? | |
| Feb 19, 2017 at 7:22 | comment | added | Adam Sheffer | Thank you nfdc23. I indeed already looked for this in the Bochnak-Coste-Roy book. While the claim is stated in Prop 3.3.14, the proof refers to Prop 3.3.2 which in turn refers to two other sources. One of these is in French and the other I didn't easily find. I'd appreciate it if you could point out what is the issue with my argument. I skipped several details in the above sketch, but I can't see which step is problematic. | |
| Feb 19, 2017 at 6:00 | comment | added | nfdc23 | See Prop. 3.3.10 and then Prop. 3.3.14 of the book Real Algebraic Geometry by Bochnak, Coste, and Roy; also see Remark 3.3.15 for a warning about failure of density in the classical topology for this algebraic notion of smoothness. I suggest that you first read the Introduction to appreciate some of the subtleties of real algebraic geometry (since your argument for hypersurfaces is not quite convincing since it seems to be mixing up what is a definition and what is a theorem, so that it seems you might not be aware of the pitfalls for passing between geometry and algebra over $\mathbf{R}$). | |
| Feb 19, 2017 at 4:53 | comment | added | JSE | I think you have to be a little careful about using Nullstellensatz over a field that's not algebraically closed. For instance, if f = x^2 + y^2, then x vanishes on every real point where f does, but x isn't in the ideal generated by (f). Now you may not think of V = V(f) as a "variety of dimension n-1," and I can see why. But this shows someo of the subtleties. | |
| Feb 19, 2017 at 3:13 | history | asked | Adam Sheffer | CC BY-SA 3.0 |