PhD Synopsis seminar by Subhajit Ghosh

Title : ON THE NUMBER OF CONNECTED COMPONENTS OF POLYNOMIAL LEMNISCATES: DETERMINISTIC AND RANDOM

Abstract :

We investigate the number of connected components of unit lemniscates associated with random and deterministic polynomials. For a monic polynomial p, we define its unit lemniscate Λ_p:= {z: |p(z)| < 1}. The maximum principle implies that each connected component of the lemniscate must contain a root of the polynomial. Therefore, the lemniscate for a polynomial of degree n, can have at most n components.

First, we investigate the number of components of a typical lemniscate. We show that the expected number of components for lemniscates with roots chosen uniformly from unit disk D, is bounded above and below by a constant multiple of √ n.On the other hand, if the roots are chosen uniformly from the unit circle S^1, the expected number of connected components, normalized by n, converges to 1.

Second, we analyze the maximum number of components when the roots of the polynomial lie within a compact set K ⊂ C with positive logarithmic capacity c(K). We prove that the quantity M(K) = lim sup n→∞ C_n(K)/n, where C_n(K) is the maximum number of connected components of Λ_p over all monic polynomial p of degree n with roots inside K, satisfies M(K) < 1 if c(K) < 1, and M(K) = 1 if c(K) ≥ 1. This resolves a question posed by Erdὂs et al. in 1958. Together, these results provide significant insights into the asymptotic behavior of the number of connected components of polynomial lemniscates.