A note on the Erd\"os minimal area problem

TL;DR

This paper addresses Erd"os, Herzog, and Piranian's question by analyzing the minimal area of polynomial lemniscates with zeros confined to a compact set K of logarithmic capacity greater than 1.

Contribution

It provides a solution to a longstanding problem regarding the minimal area of polynomial lemniscates under zero location constraints.

Findings

Determined the minimal area of polynomial lemniscates for sets with capacity > 1

Extended understanding of polynomial lemniscates in complex analysis

Resolved a question posed by Erd"os and colleagues

Abstract

We answer a question of Erd\"os, Herzog, and Piranian on the minimal area of polynomial lemniscates when all the zeros of the polynomial are constrained to lie on a compact set K whose logarithmic capacity is strictly larger than 1.