Irreducibility of lemniscates

TL;DR

This paper proves that lemniscates, defined as the set of points where the modulus of a polynomial equals one, are irreducible algebraic curves over the real numbers.

Contribution

It establishes the irreducibility of lemniscates as real algebraic curves, a result previously unproven in the literature.

Findings

Lemniscates are irreducible real algebraic curves.

The proof relies on algebraic and geometric properties of polynomial level sets.

This result impacts the understanding of the algebraic structure of polynomial lemniscates.

Abstract

We prove that lemniscates (i.e., sets of the form $|P(z)|=1$ where $P$ is a complex polynomial) are irreducible real algebraic curves.