\(p\)-Conjecture for tame automorphisms of \(\mathbb{C}^3\)

Abstract

The famous Jung-van der Kulk [4, 11] theorem says that any polynomial automorphism of \(\mathbb{C}^2\) can be decomposed into a finite number of affine automorphisms and triangular automorphisms, i.e. that any polynomial automorphism of \(\mathbb{C}^2\) is a tame automorphism. In [5] there is a conjecture saying that for any tame automorphism of \(\mathbb{C}^3,\) if \((p,d_2,d_3)\) is a multidegree of this automorphism, where \(p\) is a prime number and \(p\leq d_2\leq d_3,\) then \(p|d_2\) or \(d_3\in p\mathbb{N}+d_2\mathbb{N}.\) Up to now this conjecture is unsolved. In this note, we study this conjecture and give some results that are partial results in the direction of solving the conjecture. We also give some complimentary results.