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A note on the Erdös minimal area problem

Subhajit Ghosh, Koushik Ramachandran

Abstract

We answer a question of Erdös, 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.

A note on the Erdös minimal area problem

Abstract

We answer a question of Erdös, 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.

Paper Structure

This paper contains 3 sections, 1 theorem, 16 equations.

Table of Contents

  1. Introduction
  2. Capacity does not determine mu(K)
  3. Main theorem

Key Result

Theorem 3.1

Let $t > 1$ be fixed. Let $K\subset{\mathbb{C}}$ be a compact set with $cap(K)\geq t$. Then there exists a constant $\rho = \rho(t) > 0$ such that for all large enough $n$, we have In particular, $\mu(K) = 0$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (4)

  • Example 2.1
  • Theorem 3.1
  • Remark 3.2
  • proof