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Wild Multidegrees of the Form $(d,d_{2},d_{3})$ for Fixed $d\geq 3$

Volume 60 / 2012

Marek Karaś, Jakub Zygadło Bulletin Polish Acad. Sci. Math. 60 (2012), 211-218 MSC: Primary 14R10; Secondary 14R15. DOI: 10.4064/ba60-3-2

Abstract

Let $d$ be any integer greater than or equal to $3.$ We show that the intersection of the set $\mathop{\rm mdeg} (\mathop{\rm Aut} (\mathbb{C}^{3}))\setminus \mathop{\rm mdeg} (\mathop{\rm Tame} (\mathbb{C} ^{3}))$ with $\{(d_{1},d_{2},d_{3})\in ( \mathbb{N}_+) ^{3}:d=d_{1}\leq d_{2}\leq d_{3}\}$ has infinitely many elements, where $\mathop{\rm mdeg} h=(\mathop{\rm deg} h_1,\ldots,\mathop{\rm deg} h_n)$ denotes the multidegree of a polynomial mapping $h=(h_1,\ldots,h_n):\mathbb{C}^n\rightarrow\mathbb{C}^n.$ In other words, we show that there are infinitely many wild multidegrees of the form $(d,d_2,d_3),$ with fixed $d\geq 3$ and $d\leq d_2 \leq d_3,$ where a sequence $(d_1,\ldots,d_n)\in\mathbb{N}^n$ is a wild multidegree if there is a polynomial automorphism $F$ of $\mathbb{C}^n$ with $\mathop{\rm mdeg} F=(d_1,\ldots,d_n),$ and there is no tame automorphism of $\mathbb{C}^n$ with the same multidegree.

Authors

  • Marek KaraśInstytut Matematyki
    Wydział Matematyki i Informatyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail
  • Jakub ZygadłoInstytut Informatyki
    Wydział Matematyki i Informatyki
    Uniwersytet Jagielloński
    Łojasiewicza 6
    30-348 Kraków, Poland
    e-mail

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