Multidegrees of tame automorphisms of $\mathbb{C}^n$

Volume 477 / 2011

Marek Karaś Dissertationes Mathematicae 477 (2011), 1-55 MSC: Primary 14Rxx; Secondary 14R10. DOI: 10.4064/dm477-0-1


Let $F=(F_{1},\ldots,F_{n}) :\mathbb{C}^{n}\rightarrow \mathbb{C}^{n}$ be a polynomial mapping. By the multidegree of $F$ we mean $\mathop{\rm mdeg} % F=(\deg F_{1},\ldots,\deg F_{n}) \in \mathbb{N}^{n}.$ The aim of this paper is to study the following problem (especially for $% n=3) $: for which sequence $(d_{1},\ldots,d_{n}) \in \mathbb{N}^{n}$ is there a tame automorphism $F$ of $\mathbb{C}% ^{n}$ such that $\mathop{\rm mdeg} F=(d_{1},\ldots, d_{n})?$ In other words we investigate the set $\mathop{\rm mdeg} (\mathop{\rm Tame}% (\mathbb{C}^{n})) $, where $\mathop{\rm Tame}(\mathbb{C}% ^{n}) $ denotes the group of tame automorphisms of $\mathbb{C}^{n}.$

Since $\mathop{\rm mdeg} (\mathop{\rm Tame}( \mathbb{C}^{n})) $ is invariant under permutations of coordinates, we may focus on the set $\{ (d_{1},\ldots,d_{n}) :d_{1}\leq \cdots \leq d_{n}\} \cap \mathop{\rm mdeg} (\mathop{\rm Tame}% (\mathbb{C}^{n})).$

Obviously, we have $\{ (1,d_{2},d_{3}) :1\leq d_{2}\leq d_{3}\} \cap \mathop{\rm mdeg} (\mathop{\rm Tame}( \mathbb{C}^{3})) = \{ (1,d_{2},d_{3}) :1\leq d_{2}\leq d_{3}\}.$ Not obvious, but still easy to prove is the equality $\mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3}) )\cap \{ (2,d_{2},d_{3}) : 2\leq d_{2}\leq d_{3}\} = \{ (2,d_{2},d_{3}) :2\leq d_{2}\leq d_{3}\}.$

We give a complete description of the sets $\{ (3,d_{2},d_{3}) : 3\leq d_{2}\leq d_{3}\}\!\cap \mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3}) )$ and $\{ (5,d_{2},d_{3}) :5\leq d_{2}\leq d_{3}\} \cap \mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3})).$ In the examination of the last set the most difficult part is to prove that $(5,6,9) \notin \mathop{\rm mdeg} % (\mathop{\rm Tame}(\mathbb{C}^{3})).$ To do this, we use the two-dimensional Jacobian Conjecture (which is true for low degrees) and the Jung–van der Kulk Theorem.

As a surprising consequence of the method used in proving that $( 5,6,9) \notin \mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}% ^{3})) $, we show that the existence of a tame automorphism $F$ of $\mathbb{C}^{3}$ with $\mathop{\rm mdeg} F=(37,70,105) $ implies that the two-dimensional Jacobian Conjecture is not true.

Also, we give a complete description of the following sets: $\{ (p_{1},p_{2},d_{3}) : 2< p_{1}< p_{2}\leq d_{3}, p_{1},p_{2} \text{ prime numbers }\} \cap \mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3})) $, $\{ (d_{1},d_{2},d_{3}) : d_{1}\leq d_{2}\leq d_{3}, $ $ d_{1},d_{2}\in 2\mathbb{N}+1$, $\gcd (d_{1},d_{2}) =1\} \cap \mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3})).$ Using the description of the last set we show that $\mathop{\rm % mdeg}(\mathop{\rm Aut}(\mathbb{C}^{3})) \backslash \mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3}) ) $ is infinite.

We also obtain a (still incomplete) description of the set $\mathop{\rm mdeg} (\mathop{\rm Tame}(\mathbb{C}^{3}) ) \cap \{ (4,d_{2},d_{3}) : 4\leq d_{2}\leq d_{3}\} $ and we give complete information about $\mathop{\rm mdeg} F^{-1}$ for $% F\in \mathop{\rm Aut}(\mathbb{C}^{2}).$


  • Marek KaraśInstytut Matematyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland

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