On the number of connected components of polynomial lemniscates: Deterministic and Random

Title:       On the number of connected components of polynomial lemniscates: Deterministic and Random

Abstract:   For a complex polynomial p, its unit lemniscate is defined as the set $\Lambda_p : = \{z\in\mathbb{C}: |p(z)| < 1\}$. In this talk, we investigate the number of connected components of unit lemniscates associated with deterministic  and random polynomials. First, we study the number of components of a typical lemniscate. We show that the expected number of components of lemniscates with zeros chosen uniformly from  the unit disk $\mathbb{D}$, is bounded above and below by a constant multiple of $\sqrt{n}$. On the other hand, we show that if the zeros are chosen uniformly from the unit circle $\mathbb{S}^1$, the expected number of connected components, normalized by n, converges to 1/2. 

Second, we analyze the deterministic problem about the maximum number of components of lemniscates, when the zeros of the polynomial lie in a fixed compact set $K \subset \mathbb{C}$. Our results, which are essentially the best possible, show that this maximal number depends on whether the logarithmic capacity of the set K is less than 1, equal to 1, or greater than 1, thereby resolving a question posed by Erdős, Herzog, and Piranian in 1958.