On weighted bidegree of polynomial automorphisms of C^2

TL;DR

This paper investigates the structure of weighted degrees of polynomial automorphisms of C^2, revealing how these degrees relate to divisibility conditions and automorphism construction.

Contribution

It characterizes the set of weighted bidegrees of polynomial automorphisms of C^2, extending known degree divisibility results to weighted degrees.

Findings

Weighted degrees follow divisibility patterns similar to standard degrees.

Construction of automorphisms with prescribed weighted degrees under divisibility conditions.

Provides a detailed description of the set of possible weighted bidegrees.

Abstract

Let $F=(F_1,F_2):C^2 ---> C^2 be a polynomial automorphism. It is well know that deg F_1 | deg F_2 or deg F_2 | deg F_1. On the other hand, if (d_1,d_2) \in (N\{0})^2 is such that d_1 | d_2 or d_2 | d_1, then one can construct a polynomial automorphism F=(F_1,F_2) of C^2 with deg F_1=d_1 and deg F_2=d_2. Let us fix w=(w_1,w_2) \in (N\{0})^2 and consider the weighted degree on C[x,y] with wdeg x=w_1$ and wdeg y=w_2. In this note we address the structure of the set {(\wdeg F_1,\wdeg F_2) : (F_1,F_2) is an automorphism of C^2}.