Timeline for The rank of a perturbed triangular matrix

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Jun 27, 2017 at 14:45	history	edited	Seva	CC BY-SA 3.0	added 1065 characters in body
Jun 27, 2017 at 12:45	comment	added	Seva		@FedorPetrov: I thought about the circulant matrices in connection with this problem, without getting anywhere (which does not mean, of course, that this is a wrong direction).
Jun 27, 2017 at 11:31	comment	added	Fedor Petrov		(continuation) Say, for the cyclic group of order $n$, we need a polynomial $P(x)=c_0+c_1x+\dots+c_{n-1}x^{n-1}$ satisfying $c_0\ne 0$, $c_kc_{n-k}=0$ for $k=1,\dots,n-1$ and satisfying $P(x)=0$ for almost all (say, all but about $\sqrt{n}$ different $x$) $n$-roots of unity. Is this possible?
Jun 27, 2017 at 11:31	comment	added	Fedor Petrov		Restrict the initial question to circulant matrices in some (additive, by the way why? Non-abelian groups also look ok) group $G$: $b_{ij}=f(i-j)$, the condition becomes $f(0)\ne 0$, $f(x)f(-x)=0$ for $x\ne 0$. We want the small rank. What is the rank of a circulant matrix? It equals to the number of characters $\chi$ satisfying $\chi(\sum_{g\in G} f(g)g)\ne 0$. (to be continued)
Jun 27, 2017 at 10:01	history	answered	Seva	CC BY-SA 3.0