Let d be any integer greater than or equal to 3. We show that the intersection of the set mdeg(Aut(ℂ³))∖ mdeg(Tame(ℂ³)) with {(d₁,d₂,d₃) ∈ (ℕ ₊)³: d = d₁ ≤ d₂≤ d₃} has infinitely many elements, where mdeg h = (deg h₁,...,deg hₙ) denotes the multidegree of a polynomial mapping h = (h₁,...,hₙ): ℂⁿ → ℂⁿ. In other words, we show that there are infinitely many wild multidegrees of the form (d,d₂,d₃), with fixed d ≥ 3 and d ≤ d₂ ≤ d₃, where a sequence (d₁,...,dₙ)∈ ℕ ⁿ is a wild multidegree if there is a polynomial automorphism F of ℂⁿ with mdeg F = (d₁,...,dₙ), and there is no tame automorphism of ℂⁿ with the same multidegree.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-3-2,
author = {Marek Kara\'s and Jakub Zygad\l o},
title = {Wild Multidegrees of the Form (d,d2,d3) for Fixed d >= 3},
journal = {Bulletin of the Polish Academy of Sciences. Mathematics},
volume = {60},
year = {2012},
pages = {211-218},
zbl = {1251.14045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-3-2}
}
Marek Karaś; Jakub Zygadło. Wild Multidegrees of the Form (d,d₂,d₃) for Fixed d ≥ 3. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 60 (2012) pp. 211-218. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba60-3-2/