Tame Automorphisms Of C~3with Multidegree Of Special Form

Automorphisms of polynomial ring play an important role in afne algebric geom-etry. The famous Jacobian Conjecture is one of open problems of this field, which saysthat if the Jacobian determinant of a polynomial map over a field of characteristic0is a non-zero constant, then the polynomial map is a polynomial automorphism. Theconjecture is studied extensively. Tame generation problem is one of problems relatedto the Jacobian Conjecture. The tame automorphism of complex field is finite composi-tion of elementary automorphisms and afne automorphisms. In linear algebra, we knowthat each invertible matrix is a finite product of elementary matrices. Similarly, is eachpolynomial automorphism a finite composition of elementary automorphisms and afneautomorphisms? Or is each polynomial automorphism a tame automorphism? That isthe “Tame generation problem”. Initially, people believed that the answer is afrmative,which is the case in dimension1. In the case of dimension two, the famous Jung-van derKulk theorem gives the afrmative answer. However, in the case of dimension3, Nagata presented in1972the Nagata automorphism σ and he conjectured that a is not tame. This conjecture was not solved until Shestakov and Umirbaev proved it true in2003. Shestakov and Umirbaev put the polynomial ring into a Possion algebra and defined elementary reductions and reductions of type Ⅰ-Ⅳ by introducing the calculation of Possion bracket. They proved that a tame automorphism admits either an elementary reduction or a reduction of type Ⅰ-Ⅳ.Thanks to the work of Karas, multidegrees of polynomial automorphisms play an es-sential role in the research of tame automorphism with three variables. Kuroda has proved that there exists no tame automorphism admitting a reduction of type IV. Karas proved that there exist infinite number of multidegrees of nontame automorphisms. Though many jobs have been done, the description of tame automorphism group is not clear. In this paper, we proved a class of automorphisms with multidegree of some special form are not tame automorphisms. The main results of this paper are as follows.Theorem3.2.1Let d1≤d2≤d3be positive integers, gcd(d1, d2)=1,2d2≥d1+d3, and d1<d3-d2<2d1. Then (d1, d2,d3) is a multidegree of some tame automorphism if and only if d1|d3.Theorem3.2.2Let d1≤d2≤d3be positive integers, gcd(d1, d2)=1,2d2> d1+d3, and d3-d2<d1.(1) If d1is odd, then (d1,d2, d3) is a multidegree of some tame automorphism if and only if d1|d3.(2) If d1is even, such that d1(?)2d3, then (d1, d2, d3) is not a multidegree of any tame automorphism.Theorem3.3.1Let a, k be positive integers, a≥5, gcd(a, k)=1, and (a, a+2k, a+3k)≠(6,8,9). Then (a, a+2k, a+3k) is not a multidegree of a tame automorphism. Theorem3.3.2Let d1, d2be odd numbers, gcd(d1, d2)=1. If (d1, d2, d3) is not amultidegree of a tame automorphism, then (rd1, rd2, rd3) is not a multidegree of a tameautomorphism for arbitrary positive integer r.