Wild Multidegrees of the Form (d, d_2, d_3) for Fixed d ≥ 3

摘要

Let d be any integer greater than or equal to 3. We show that the intersection of the set mdeg(Aut(C~3))mdeg(Tame(C~3)) with {{d_1,d_2, d_3) ∈ (N+)~3:d = d_1 ≤ d_2 ≤ d_3} has infinitely many elements, where mdeg h = (deg h_1,..., degh_n) denotes the multidegree of a polynomial mapping h = (h_1,..., h_n): C~n → 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 ≥ 3 and d ≤ d_2 ≤ d_3, where a sequence (d_1,..., d_n) ∈ N~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 automorphism of C~n with the same multidegree.