Timeline for Partitions of $\mathbb{R}^d$ by implicit polynomial equations
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 10, 2012 at 1:32 | comment | added | Igor Rivin | In fact, looking at the Milnor paper, that seems to be exactly what he does (jiggle the $\epsilon$ a bit...) | |
| Sep 9, 2012 at 12:04 | comment | added | Saugata Basu | It could happen of course that $P>0$ is empty, even though $P\geq 0$ is not. However, the reduction mentioned above works to give the same bound. If one is interested in bounding the number of connected components of the basic semi-algebraic defined by $P_1 > 0,\ldots,P_s >0$, then a slightly better bound (better dependence on $s$) is available in the case where the inequalities are strict. Need to thank Patricia Hersh for pointing me to this discussion since I don't follow this forum regularly. | |
| Sep 9, 2012 at 4:52 | comment | added | Will Sawin | The number of connected components of $P> 0$ is the same as the number for $P\geq \epsilon$ for $\epsilon$ sufficiently small. This doesn't work as well for two-sided bounds but if it's one-sided I don't think that's a big problem | |
| Sep 9, 2012 at 2:07 | comment | added | Igor Rivin | Or is the "semi-algebraically connected component" in fact the open set? | |
| Sep 9, 2012 at 2:00 | comment | added | Igor Rivin | Welcome to MO! Perhaps you can address @Patricia's question: The Oleinik/Milnor/Thom bound is for the number of connected components of $P\geq 0.$ Does anything particularly unpleasant happen if the inequalities are strict? | |
| Sep 9, 2012 at 1:47 | history | answered | Saugata Basu | CC BY-SA 3.0 |