On automorphisms of quasi-circular domains fixing the origin

TL;DR

This paper establishes degree bounds for automorphisms of quasi-circular domains fixing the origin, generalizing Cartan's theorem from circular to quasi-circular domains using resonance concepts.

Contribution

Introduces resonance and quasi-resonance orders to uniformly bound the degree of polynomial automorphisms of quasi-circular domains.

Findings

Provides a uniform upper bound for automorphism degrees

Generalizes Cartan's theorem to quasi-circular domains

Shows automorphisms are polynomial mappings

Abstract

It is known that automorphisms of quasi-circular domains fixing the origin are polynomial mappings. By introducing the so-called resonance order and quasi-resonance order, we provide a uniform upper bound for the degree of such polynomial automorphisms. As a particular consequence of our result, we obtain a generalization of the classical Cartan's theorem for circular domains to the quasi-circular case.