# On the degree of automorphisms of quasi-circular domains fixing the origin

## TL;DR

This paper establishes a precise condition linking the degree of automorphisms of quasi-circular domains fixing the origin to their resonance order, resolving a previously conjectured relationship.

## Contribution

It provides a necessary and sufficient condition using Bergman representative coordinates, confirming a conjecture about automorphism degrees in quasi-circular domains.

## Key findings

- Degree of automorphisms equals resonance order under specified conditions
- Bergman representative coordinates are key to characterizing automorphism degrees
- Conjecture on automorphism degree has been proven

## Abstract

By using the Bergman representative coordinates, we give the necessary and sufficient condition for the degree of automorphisms of quasi-circular domains fixing the origin to be equal to the resonance order, thus solving a conjecture of the author.