Wild multidegrees of the form (d,d_2,d_3) for given d greather than or equal to 3

TL;DR

This paper demonstrates that for any fixed d ≥ 3, there are infinitely many wild multidegrees of the form (d, d_2, d_3) with d ≤ d_2 ≤ d_3, highlighting the complexity of polynomial automorphisms in three variables.

Contribution

It establishes the existence of infinitely many wild multidegrees of a specific form for polynomial automorphisms in three variables, expanding understanding of wild automorphisms.

Findings

Infinitely many wild multidegrees of the form (d, d_2, d_3) for fixed d ≥ 3

Existence of multidegrees not realizable by tame automorphisms

Characterization of wild multidegrees with fixed first component

Abstract

Let d be any number greather than or equal to 3. We show that the intersection of the set mdeg(Aut(C^3))\ mdeg(Tame(C3)) with {(d_1,d_2,d_3) : d=d_1 =< d_2 =< d_3} has infinitely many elements, where mdeg h = (deg h_1,...,deg h_n) denotes the multidegree of a polynomial mapping h=(h_1,...,h_n):C^n ---> C^n. In other words, we show that there is infiniltely many wild multidegrees of the form (d,d_2,d_3), with fixed d >= 3 and d =< d_2 =< d_3, where a sequences (d_1,...,d_n) is a wild multidegree if there is a polynomial automorphism F of C}^n with mdeg F=(d_1,...,d_n), and there is no tame autmorphim of C^n with the same multidegree.