5 problems

Let $P$ be a complex polynomial and let $a$ and $b$ satisfy … Assume that all roots of $P$ other than $a$ and $b$ are simple. Definiteness conjecture. Then $P$ is definite. The con…

Let $p=P'$ and $q=Q'$ be polynomials, and let the polynomial moments (1.2) be the moments associated with $P$ and $q$. The Polynomial Composition condition (PCC) is that there exis…

Let $X=(X_1,\dots,X_n)$ be a jointly Gaussian random vector, let $P(x)\in\mathbb{C}[x]$, and let $\operatorname{O}_n(\mathbb{C})$ act by linear changes of variables. A one-paramete…

Let $X=(X_1,\dots,X_n)$ be a random vector with independent centered, variance-one Gaussian components. Let … A subspace $V$ is an MZ space if $f^m\in V$ for all $m\geq1$ implies…

Let $X=(X_1,\dots,X_n)$ be a random vector with a multivariate normal distribution, and let $P(x_1,\dots,x_n)\in \mathbb{C}[x_1,\dots,x_n]$ be a complex-valued polynomial. Gaussian…