On biholomorphisms between bounded quasi-Reinhardt domains

TL;DR

This paper introduces quasi-Reinhardt domains, proves biholomorphisms fixing the origin are polynomial, and provides a degree bound, extending Cartan's linearity theorem to these domains.

Contribution

It generalizes Cartan's linearity theorem to quasi-Reinhardt domains and establishes polynomial nature and degree bounds for biholomorphisms fixing the origin.

Findings

Biholomorphisms are polynomial mappings

Degree of polynomial mappings has a uniform upper bound

Generalization of Cartan's theorem to quasi-Reinhardt domains

Abstract

In this paper, we define what is called a quasi-Reinhardt domain and study biholomorphisms between such domains. We show that all biholomorphisms between two bounded quasi-Reinhardt domains fixing the origin are polynomial mappings, and we give a uniform upper bound for the degree of such polynomial mappings. In particular, we generalize the classical Cartan's linearity theorem for circular domains to quasi-Reinhardt domains.