On origin-preserving automorphisms of quasi-circular domains (Q1651391)

In [Manuscr. Math. 3, 257--270 (1970; Zbl 0202.36504)], \textit{W. Kaup} showed that every origin-preserving automorphism of a quasi-circular domain is a polynomial mapping. In [Bull. Sci. Math. 140, No. 1, 92--98 (2016; Zbl 1338.32020)], the reviewer gave a uniform upper bound for such polynomial automorphisms, in terms of the so-called ``quasi-resonance order'', and conjectured that the optimal uniform upper bound should be given by the generally smaller ``resonance order''. In the paper under review, the authors classify all possible degrees for origin-preserving automorphisms of quasi-circular domains in dimension two, by using the Bergman representative coordinates. As an application, they give a counterexample to the above mentioned conjecture of the reviewer [loc. cit.], and at the same time confirm that the upper bound in terms of the quasi-resonance order given by the reviewer [loc. cit.] is indeed optimal.