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2 Bulletin of the Polish Academy of Sciences. Mathematics

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On weighted bidegree of polynomial automorphisms of C2

Karaś Marek

Bulletin of the Polish Academy of Sciences. Mathematics

2022

Vol. 70, no. 2

107--114

Let F=(F1,F2):C2→C2 be a polynomial automorphism. It is well known that degF1|degF2 or degF2|degF1. On the other hand, if (d1,d2)∈N2+=(N∖{0})2 is such that d1|d2 or d2|d1, then one can construct a polynomial automorphism F=(F1,F2) of C2 with degF1=d1 and degF2=d2. Let us fix w=(w1,w2)∈N2+ and consider the weighted degree on C[x,y] with degwx=w1 and degwy=w2. In this note we address the structure of the set {(degwF1,degwF2):(F1,F2) is an automorphism of C2}. This is a very first, but necessary, step in studying weighted multidegrees of polynomial automorphisms.

Wild Multidegrees of the Form (d,d2,d3) for Fixed d≥3

Karaś M., Zygadło J.

Bulletin of the Polish Academy of Sciences. Mathematics

2012

Vol. 60, no 3

211--218

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 mdegh=(degh1,…,deghn) 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)∈Nn is a wild multidegree if there is a polynomial automorphism F of Cn with mdegF=(d1,…,dn), and there is no tame automorphism of Cn with the same multidegree.