[Paper Review] Tame automorphisms with multidegrees in the form of arithmetic progressions

Jiantao Li, Xiankun Du|arXiv (Cornell University)|Dec 28, 2011

Advanced Differential Equations and Dynamical Systems7 references3 citations

TL;DR

This paper classifies which arithmetic progression multidegrees $(a, a+d, a+2d)$ can arise from tame automorphisms of $\mathbb{C}^3$. It proves that such multidegrees are realizable if and only if $a \mid 2d$, except for a single exceptional family $(4i, 4i+ij, 4i+2ij)$ with odd $j$, whose realizability depends on a conjecture about Poisson bracket degrees.

ABSTRACT

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 $a mid 2d$, 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, $(a, a+d, a+2d) otin\mdeg(\Tame(\mathbb{C}^3))$. 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.

Motivation & Objective

• To determine which arithmetic progression multidegrees $(a, a+d, a+2d)$ can be realized by tame automorphisms of $\mathbb{C}^3$.
• To resolve the realizability of multidegrees in the exceptional case $(4i, 4i+ij, 4i+2ij)$ with odd $j$, which remains open.
• To connect the realizability of these multidegrees to a conjecture on the lower bound of Poisson bracket degrees.
• To extend previous results on multidegrees of tame automorphisms in dimension three, particularly for symmetric and consecutive integer sequences.

Proposed method

• Use of the $*$-reduced pair inequality from Shestakov and Umirbaev's work on the Nagata automorphism to bound the degree of polynomial compositions.
• Application of Theorem 2.1 to derive lower bounds on $\deg g(F_1, F_2)$ when $F$ admits a reduction, based on the Poisson bracket $[F_1, F_2]$.
• Analysis of elementary reductions via permutation of variables and degree comparison, showing contradictions when $a \nmid 2d$ and in the exceptional case.
• Reduction to the study of $\deg[F_i, F_j]$ for pairs of polynomials, particularly $[F_1, F_3]$ in the exceptional case.
• Use of the conjecture by Jie-tai Yu on the lower bound of $\deg[f,g]$ for algebraically independent polynomials with dependent leading forms.
• Systematic case analysis of all three types of elementary reductions: on $F_1$, $F_2$, and $F_3$, showing none are possible in the exceptional case unless the conjecture holds.

Experimental results

Research questions

• RQ1 For which positive integers $a$ and $d$ is the multidegree $(a, a+d, a+2d)$ in $\mathrm{mdeg}(\mathrm{Tame}(\mathbb{C}^3))$?
• RQ2 What is the role of the condition $a \mid 2d$ in determining the realizability of such multidegrees?
• RQ3 Why is the family $(4i, 4i+ij, 4i+2ij)$ with odd $j$ the only exceptional case not covered by the $a \mid 2d$ criterion?
• RQ4 How does the conjecture of Jie-tai Yu on Poisson bracket degrees relate to the realizability of these multidegrees?
• RQ5 Can the multidegree $(8,10,12)$ be realized by a tame automorphism of $\mathbb{C}^3$?

Key findings

• If $a \mid 2d$, then the multidegree $(a, a+d, a+2d)$ is in $\mathrm{mdeg}(\mathrm{Tame}(\mathbb{C}^3))$.
• If $a \nmid 2d$, then $(a, a+d, a+2d) \notin \mathrm{mdeg}(\mathrm{Tame}(\mathbb{C}^3))$ except possibly for the family $(4i, 4i+ij, 4i+2ij)$ with $i,j \in \mathbb{N}$ and $j$ odd.
• The exceptional case $(4i, 4i+ij, 4i+2ij)$ with odd $j$ cannot be realized unless the conjecture of Jie-tai Yu on Poisson bracket degrees holds.
• The multidegree $(4,5,6)$ is not realizable, as previously shown by Karaś, and this is consistent with the $a \nmid 2d$ condition.
• The only unresolved case is $(8,10,12)$, which corresponds to $i=2$, $j=1$ in the exceptional family.
• For continuous odd integers, only $(1,3,5)$ is realizable among such sequences, as $d_1=1$ is the only case satisfying $a \mid 2d$.