Title: Number of connected components of polynomial lemniscates

Abstract:  A lemniscate of a \emph{monic} complex polynomial \(p\) is a sublevel set of its modulus, namely
\[
\Lambda(p):=\{z \in \mathbb{C}: |p(z)| < 1\}.
\]
The study of lemniscates was pioneered by Erd\H{o}s, Herzog, and Piranian in 1958, where they posed several extremal questions concerning the geometric and topological properties of lemniscates.


In this talk, we explore the problem concerning the maximum possible number of connected components of a lemniscate. Without any constraint on the location of the roots, for a polynomial of degree $n$, the number of components can range between \(1\) to \(n\). However, if the roots lie in a fixed compact set \(K\subset\mathbb{C}\), can the number of components still be as large as the degree?

Let \(c(K)\) denote the logarithmic capacity of a compact set \(K\subset\mathbb C\). 
For \(n\geq 1\), let \(\mathscr C_n(K)\) be the maximum number of connected components of
\(\Lambda_p\), taken over all monic polynomials \(p\) of degree \(n\) whose roots lie in \(K\). 
We prove that for all \emph{Connected} sets, 
\[
\boxed{
\begin{aligned}
    M(K)&<1,  &&\text{if } c(K)<1,\\
    M(K)&=1,  &&\text{if } c(K)> 1.
\end{aligned}
}
\]
Where 
$M(K) = \limsup_{n \to \infty} \frac{\mathscr{C}_n(K)}{n}$. We conclude the talk with an approximate solution to the critical case \(c(K)=1\). 
We show that if \(K\) is a Jordan domain with sufficiently regular boundary, the $M(K)=1.$ This talk is based on joint work with Koushik Ramachandran.