6 problems

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…

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 $f_i$.

Let $p$ be a polynomial of degree $n$ with $p(0)=0$ and $p'(0)\neq0$. Smale's conjecture. At least one root $\mu_j$ of $p'$ satisfies … The source lists this among open conjectures…

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…