8 problems

Let $n \geq 1$, and let $p\colon\mathbb{C}\to\mathbb{C}$ be a monic polynomial of degree $n$. For $R\geq 0$, define … and … The Erdős–Herzog–Piranian conjecture. If…

Consider configurations of zeros used to form monic degree-$n$ polynomials and their lemniscates $\Lambda_p=\{z\in\mathbb C:|p(z)|<1\}$. The numerical minimizers are configurations…

Let $p$ be a polynomial of degree $n$, let $\Lambda_p=\{z\in\mathbb C:|p(z)|<1\}$, let $m(\Lambda_p)$ be its area, and let $\rho(\Lambda_p)$ be its inradius. Solynin–Williams conje…

Let $p$ range over monic polynomials of degree $n$ whose zeros lie in the closed unit disc, and let $\Lambda_p=\{z\in\mathbb C:|p(z)|<1\}$ with area denoted by $A(\Lambda_p)$. Erdő…

Let $K$ be a compact subset of the complex plane with logarithmic capacity $c(K)=1$. Let $M(K)$ denote the maximal number of connected components of a polynomial lemniscate associa…

Erdős's conjecture. The curve $C_{1,k}$ is the longest curve of the form $P^{-1}(S^1)$ among monic polynomials $P\in\mathbf{C}[T]$ of degree $k$. The paper's computation is intende…

For $r>0$, let $M_r$ be the collection of polynomials for which the minimum distance between any two distinct zeros of $p$ is at least $r$ times the diameter of the zero set of $p$…

Let $E(f)=\{z\in\mathbb{C}:|f(z)|\leq 1\}$ be the filled-in lemniscate of a polynomial $f$, and let a set have transfinite diameter $1$. Consider monic polynomials $f_d$ of degree…