Timeline for Partitions of $\mathbb{R}^d$ by implicit polynomial equations
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 8, 2012 at 15:40 | history | edited | Igor Rivin | CC BY-SA 3.0 |
added the precise result
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| Sep 7, 2012 at 18:47 | comment | added | Patricia Hersh | @Igor: thanks for explaining -- I didn't have a good intuition for how much this would drop the number of components. | |
| Sep 7, 2012 at 15:42 | comment | added | Joseph O'Rourke | @Igor: Very useful to know the asymptotic growth rate. (I should have thought of Basu's work myself.) Thanks! | |
| Sep 7, 2012 at 14:36 | comment | added | Igor Rivin | @Patricia: No, I don't think this reduces the number of connected components, which should be equal to the number of components of $P\geq 0$ plus the number for $P\leq 0$ but I am not 100% certain -- it is conceivable that one has two components meeting at a point, or some lower dimensional stratum -- presumably one can bound the number of such occurrences but a lower order term than the main term in the bound... | |
| Sep 7, 2012 at 13:57 | comment | added | Patricia Hersh | @Igor: good point. My real concern was that throwing in $P=0$ as your statement does might reduce the number of connected components. I also wondered if the fact that Joseph is actually considering a Zariski open set might mean that there are also relevant results in algebraic geometry. | |
| Sep 7, 2012 at 13:28 | comment | added | Igor Rivin | @Patricia: I am sure he is, but once you have $O(),$ what's a factor of two among friends... | |
| Sep 7, 2012 at 3:08 | comment | added | Patricia Hersh | I think Joseph is asking for the semi algebraic set that is the union of $P>0$ and $P<0$. Nonetheless, I was also wondering if work of Basu would be the place to look. | |
| Sep 7, 2012 at 2:46 | history | answered | Igor Rivin | CC BY-SA 3.0 |