Tame automorphisms with multidegrees in the form of arithmetic progressions

Details Tame automorphisms with multidegrees in the form of arithmetic progressions Tame automorphisms with multidegrees in the form of arithmetic progressions

2011-12-28

arxiv.org/abs/1112.6071v1

Mathematics

Commutative Algebra

Scientific paper

Let $(a,a+d,a+2d)$ be an arithmetic progression of positive integers. The following statements are proved: (1) If $a\mid 2d$, then $(a, a+d, a+2d)\in\mdeg(\Tame(\mathbb{C}^3))$. (2) If $a\nmid 2d$, then, except for arithmetic progressions of the form $(4i,4i+ij,4i+2ij)$ with $i,j \in\mathbb{N}$ and $j$ is an odd number, $(a, a+d, a+2d)\notin\mdeg(\Tame(\mathbb{C}^3))$. We also related the exceptional unknown case to a conjecture of Jie-tai Yu, which concerns with the lower bound of the degree of the Poisson bracket of two polynomials.

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