6 problems

SOS Conjecture. If $A(z,\bar{z})\|z\|^2\in {\rm{SOS}}_n$, then either

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 $S_n$ be the set of monic complex polynomials of degree $n\ge 2$ whose zeros lie in the closed unit disk, and let … A polynomial is locally maximal for Sendov's conjecture if i…

Let $\mu_0$ be a rotationally invariant, compactly supported probability measure satisfying some mild assumptions, and let $P^N_0$ be any sequence of polynomials whose empirical ro…

Existence conjecture for polynomial flows with prescribed heteroclinic regions. For all $n,m\in\mathbb{N}$, there exists a complex polynomial $F_{n,m}$ of degree $n+m$ such that th…

Let $p\in\mathbb{N}^*$ and let $P,Q\in\mathbb{C}[X]$ have degree $p$. Derivative-convolution Gauss–Lucas conjecture. The roots of … lie in the convex hull of the roots of $P$ and…