Multidegrees of tame automorphisms of Cn
Abstract
Let F=(F1,...,Fn):Cn --> Cn be a polynomial mapping. By the multidegree of the mapping F we mean mdeg F=(deg F1,...,deg Fn), an element of Nn. The aim of this paper is to study the following problem (especially for n=3): for which sequence (d1,...,dn) in Nn there is a tame automorphism F of Cn such that mdeg F=(d1,...,dn). In other words we investigate the set mdeg(Tame(Cn)), where Tame(Cn) denotes the group of tame automorphisms of Cn and mdeg denotes the mapping from the set of polynomial endomorphisms of Cn into the set Nn. Since for all permutation s of 1,...,n we have (d1,...,dn) is in mdeg(Tame(Cn)) if and only if (ds(1),...,ds(n)) is in mdeg(Tame(Cn)) we may focus on the set mdeg(Tame(Cn)) intersected with (d1,...,dn) : d1<=...<=dn. In the paper, among other things, we give complete description of the sets: mdeg(Tame(Cn)) intersected with (3,d2,d3):3<=d2<=d3, mdeg(Tame(Cn)) intersected with (5,d2,d3):5<=d2<=d3, In the examination of the last set the most difficult part is to prove that (5,6,9) is not in mdeg(Tame(Cn)). As a surprising consequence of the method used in proving that (5,6,9) is not in mdeg(Tame(Cn)), we obtain the result saying that the existence of tame automorphism F of C3 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(Cn)) intersected with (p1,p2,d3):3<=p1<p2<=d3, where p1 and p2 are prime numbers, mdeg(Tame(Cn)) intersected with (d1,d2,d3):d1<p2<=d3, where d1 and d2 are odd numbers such that gcd(d1,d2)=1. Using description of the last set we show that the set mdeg(Aute(Cn))(Tame(Cn)) is infinite.
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