Timeline for Integral representation of the complex homogeneous polynomial $z_1\cdots z_n$
Current License: CC BY-SA 3.0
6 events
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| Dec 1, 2015 at 2:08 | comment | added | Suvrit | Thanks for your comments! In fact, after your comment, I tried playing with these values, and constructed something with not such a nice value of $c$, and I got thinking about trying to get "optimal" choices of $q$ and $\mu$ --- though then we enter the domain of this being an open problem, I guess... | |
| Dec 1, 2015 at 1:23 | comment | added | Mike Jury | If it's just a matter of finding some $q$ and $\mu$ that work, than the requirement that $\mu$ be a probability measure doesn't matter since one can pass constants back and forth between $q$ and $\mu$, so any circularly symmetric $\mu$ with $c:=\int_{\Delta_n} |\zeta_1\cdots \zeta_n|^2 d\mu(\zeta)>0$ works, for $q(\zeta) = n!c^{-1} \zeta_1\cdots \zeta_n$. Or is your concern to find an "optimal" pair $q,\mu$ (as mentioned further down in the remark)? | |
| Dec 1, 2015 at 0:15 | comment | added | Suvrit | Seems like what I wrote will conflict with $\mu$ being a probability measure on $\Delta_n$; it seems that perhaps a choice other than what is given by Remark 5.2 may lead to "better" measures. | |
| Nov 30, 2015 at 22:27 | comment | added | Suvrit | One second thoughts, it seems that what you wrote is truly all what I need. I was overthinking the "none of the moments vanish" part. So the task reduces to constructing a circularly symmetric measure (which should also be a probability measure on $\Delta_n$?) for which $\int_{\Delta_n}|\zeta_1\cdots\zeta_n|^2d\mu(\zeta)=1/n!$ | |
| Nov 30, 2015 at 19:54 | comment | added | Mike Jury | Doesn't this follow from Remark 5.2 of that paper? All one needs to do is take a circularly symmetric $\mu$; any such $\mu$ for which $\int_{\Delta_n} |\zeta_1\cdots\zeta_n|^2 d\mu(\zeta)\neq 0$ will work (with $q(\zeta)$ a constant times $\zeta_1\cdots \zeta_n$). | |
| Nov 30, 2015 at 2:03 | history | asked | Suvrit | CC BY-SA 3.0 |