8 problems

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 $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Let $\beta\in[0,1]$, let $S(\beta)$ be the set of polynomials of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

For $n\ge2$, let $r_n(\beta)$ be the maximum of $d(P,\beta)$ over $P\in S(\beta)$ of degree $n$, and let $c_n$ denote the corresponding sharp quadratic-refinement coefficient in th…

Let $\beta$ be a real number in $[0,1]$. For a polynomial $P$ of degree at least $2$ with complex coefficients, all roots in the closed unit disk, and at least one root at $\beta$,…

Minimal-support variance conjecture.

Schmeisser's conjecture. For every $\zeta\in\operatorname{conv}(V(F))$, the closed disk centered at $\zeta$ with radius $\sigma_\infty(F)$ contains a critical point of $F$.