\(p\)-Conjecture for tame automorphisms of \(\mathbb{C}^3\)
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 conj...
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| Date: | 2025 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Lugansk National Taras Shevchenko University
2025
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| Subjects: | |
| Online Access: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2349 |
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| Journal Title: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| Summary: | 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. |
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