Timeline for Angular distribution of zero sets of sparse polynomials
Current License: CC BY-SA 3.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 1, 2015 at 18:53 | vote | accept | Vesselin Dimitrov | ||
| Nov 1, 2015 at 15:51 | answer | added | fedja | timeline score: 4 | |
| Oct 3, 2015 at 20:12 | comment | added | Vesselin Dimitrov | @IgorRivin: The one in Limit distribution of small points on algebraic tori (Duke Math. J. vol. 89, 1997.) One extension of that result was obtained by Rumely in this paper: alpha.math.uga.edu/~rr/BiluThm.pdf . | |
| Oct 3, 2015 at 20:06 | comment | added | Igor Rivin | Which result of Bilu? I could not find it by cursory mathscinet search... | |
| Oct 3, 2015 at 18:38 | comment | added | Vesselin Dimitrov | Also, if we take the number of monomials as an algebraic measure of complexity (being the number of algebraic operations needed to write down the polynomial), it could be compared to Mahler's arithmetic complexity measure $m(f) := \int_{S^1} \log{|f|} \, d\theta/2\pi$ of an irreducible integer polynomial $f \in \mathbb{Z}[x]$ (which refines / is majorized by $\log{\sum |a_i|}$). For the latter, we know by Bilu that the condition guaranteeing the equidistribution of the zeros is precisely $m(f) = o(\deg{f})$. | |
| Oct 3, 2015 at 18:00 | comment | added | Vesselin Dimitrov | There was also a Bourbaki Seminar exposition from by Khovanskii from the 1980s, on the subject of simple topology of $\mathbb{R}$-solutions and angular equidistribution of $\mathbb{C}$-solutions of a system of fewnomial equations, but I do not seem to find it on numdam. | |
| Oct 3, 2015 at 17:58 | comment | added | Vesselin Dimitrov | @IgorRivin: My understanding is that Theorem 1 of Chapter III, 3.13 (of the AMS translated edition) contains, in the special case $n = 1$, an explicit lower bound on the number $k$ of monomials that comprise a polynomial whose zeros set is not equidistributed in angle -- viz., the computable function $\varphi(k,1)$ must not be too small in such a situation, giving some lower bound on $k$, though obviously not nearly as strong as $c \deg{f}$. On the other hand I am not seeing any obvious way to improve over the trivial construction; hence the question. | |
| Oct 3, 2015 at 17:24 | comment | added | Igor Rivin | Do you have a more specific reference in Khovansky's book? | |
| Oct 3, 2015 at 2:55 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
deleted 8 characters in body
|
| Oct 3, 2015 at 2:49 | history | edited | Vesselin Dimitrov | CC BY-SA 3.0 |
deleted 275 characters in body
|
| Oct 3, 2015 at 2:39 | history | asked | Vesselin Dimitrov | CC BY-SA 3.0 |