Timeline for Interesting behaviour of Brion's formula under a degenerate change of variables
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 12, 2013 at 1:22 | comment | added | Igor Makhlin | May I then be glad that at least one person understands my question? | |
| Jul 12, 2013 at 1:10 | comment | added | Igor Makhlin | Alright then! Also, your post is exactly what's behind the words "gives me the feeling that the rest should be obvious via some formal argument" in update 2. | |
| Jul 12, 2013 at 0:13 | comment | added | Sinai Robins | Aha, thanks for the clarification, this makes the definition of your $F$ clearer and indeed it is different than the geometry that came to mind from simply projecting down the polytope and comparing the relevant tangent cones. | |
| Jul 11, 2013 at 23:52 | comment | added | Igor Makhlin | Maybe, I didn't make clear enough what $F$ is? The definition I give is a bit formal as compared to natural. The natural way might be to first introduce $F$, substituting each variable with a Laurent monomial in some other (smaller) set of variables. Then the matrix $\varphi$ can be defined via $F$. | |
| Jul 11, 2013 at 23:42 | comment | added | Igor Makhlin | Example, just in case. $C=\{(a\ge 0,b\ge 0)\}$ and $\varphi:(a,b)\rightarrow (a-b)$. Then $\varphi(C)$ is the line, $f(\varphi(C))=0.$ However, $f(C)=\dfrac{1}{(1-x)(1-y)}$ and $F(f(C))=\dfrac{1}{(1-t)(1-t^{-1})}\not\equiv 0$. (Since $\varphi=(1\text{ }-1)$, so $F$ substitutes $x$ by $t$ and $y$ by $t^{-1}$) | |
| Jul 11, 2013 at 23:25 | comment | added | Igor Makhlin | In case I have understood you correctly, this is the argument I first comforted myself with, but now believe it to be inaccurate. The reason is that, in general, $f(\varphi(C))$ is not $F(f(C))$, where $C$ is a cone, $f$ denotes integer point transform and $\varphi$ and $F$ are from the question. | |
| Jul 11, 2013 at 20:06 | history | answered | Sinai Robins | CC BY-SA 3.0 |