Multidegrees of tame automorphisms of Cn

Abstract

Let F=(F_1,...,F_n):C^n --> C^n be a polynomial mapping. By the multidegree of the mapping F we mean mdeg F=(deg F_1,...,deg F_n), an element of N^n. The aim of this paper is to study the following problem (especially for n=3): for which sequence (d_1,...,d_n) in N^n there is a tame automorphism F of C^n such that mdeg F=(d_1,...,d_n). In other words we investigate the set mdeg(Tame(C^n)), where Tame(C^n) denotes the group of tame automorphisms of C^n and mdeg denotes the mapping from the set of polynomial endomorphisms of C^n into the set N^n. Since for all permutation s of {1,...,n} we have (d_1,...,d_n) is in mdeg(Tame(C^n)) if and only if (d_s(1),...,d_s(n)) is in mdeg(Tame(C^n)) we may focus on the set mdeg(Tame(C^n)) intersected with {(d_1,...,d_n) : d_1<=...<=d_n}. In the paper, among other things, we give complete description of the sets: mdeg(Tame(C^n)) intersected with {(3,d_2,d_3):3<=d_2<=d_3}}, mdeg(Tame(C^n)) intersected with {(5,d_2,d_3):5<=d_2<=d_3}}, In the examination of the last set the most difficult part is to prove that (5,6,9) is not in mdeg(Tame(C^n)). As a surprising consequence of the method used in proving that (5,6,9) is not in mdeg(Tame(C^n)), we obtain the result saying that the existence of tame automorphism F of C^3 with mdeg F=(37,70,105) implies that two dimensional Jacobian Conjecture is not true. Also, we give the complete description of the following sets: mdeg(Tame(C^n)) intersected with {(p_1,p_2,d_3):3<=p_1<p_2<=d_3}}, where p_1 and p_2 are prime numbers, mdeg(Tame(C^n)) intersected with {(d_1,d_2,d_3):d_1<p_2<=d_3}}, where d_1 and d_2 are odd numbers such that gcd(d_1,d_2)=1. Using description of the last set we show that the set mdeg(Aute(C^n))\mdeg(Tame(C^n)) is infinite.