9 problems

Sendov's conjecture. The closed disk

Upper-envelope switching-point conjecture. The function $g$ has at most $d(n-1)$ points at which it switches between a pair of the polynomials $f_i$.

Polynomial plank conjecture. There exists a point $p\in B^d$ such that, for every $k=1,\ldots,N$, the distance from $p$ to the zero set of $P_k$ is at least $\delta_k$.

Polynomial zero-avoidance conjecture. There exists a point $p\in B^d\subset\mathbb R^d$ such that, for every $k=1,\ldots,N$, the point $p$ is at distance at least $\delta_k$ from t…

For a bounded convex set $K\subset\mathbb{C}$, an $epsilon>0$, and a polynomial $p$, let $K_epsilon$ be the $epsilon$-neighborhood of $K$ and let $Z(p,K)$ denote the number of zero…

Let $C\in M_n$ be a circulant matrix with $\lVert C\rVert\leq1$, let $\lambda$ be an eigenvalue of $C$, and let $C'$ be the principal submatrix obtained by deleting the first row a…

Let $\lambda_1,\ldots,\lambda_n$ belong to the closed unit disk, let $D=\operatorname{diag}(\lambda_1-\lambda_n,\ldots,\lambda_{n-1}-\lambda_n)$, and let $I$ and $J$ be respectivel…

The polynomial-field conjecture. There exists $n(d,k)$ such that, for every section of $\Sigma^d(\gamma_n^k)$ over $G_n^k$ with $n\geq n(d,k)$, there exists $V\in G_n^k$ such that…

Milman's conjecture. The quantity $n(d,k)$ is of order $k^d$.