Tame Automorphisms Of Polynomial Algebras

The research of automorphisms of polynomial algebras is crucial to affine algebraic geometry, which mainly studies affine spaces. Tame automorphisms constitute an important class of polynomial automorphisms. In recent years people do a lot of researches on this kind of polynomial automorphisms.Let k be a field and let k[X]= k[X1,...,Xn] be the polynomial algebra in variables X1,...,Xn over k. A polynomial map is a map of the form F= (F1,..., Fn):kn→kn, (x1,...,xn)→(F1(x1,...,xn),...,Fn(x1,...,xn)), where each Fi belongs to k[X]. It is well known that polynomial maps from k" to k" corre-spond one-to-one with k-endmorphisms of the polynomial algebra k[X]. We will not distin-guish the polynomial maps from k-endmorphisms of the polynomial algebra. A polynomial automorphism of the form E= (X1,..., Xi-1, Xi+a, Xi+1,...,Xn)is called elementary, where a is a polynomial which does not contain Xi. A polynomial automorphisms of degree one is called affine. A finite composition of elementary automorphisms and affine automorphisms is called tame. The k-automorphisms group of k[X] is denoted by Autkk[X], the group consist-ing of all tame automorphisms in dimension n is denoted by T(k, n), and the group consisting of all affine automorphisms is denoted by Affk(k, n).From linear algebra one knows that every invertible linear map over a field is a finite composition of elementary maps. For invertible polynomial maps one also has a natural question:is every invertible polynomial map over a field a finite composition of elementary maps? This problem is most widely known as the Tame Generators Problem.Since 1930s, people have done a lot of researches on the Tame Generators Problem. The classical Jung-van der Kulk theorem asserts that the Tame Generators Problem has an affirmative answer in dimension 2.Theorem 3.1 (Jung, van der Kulk) If k is a field, then Autkk[X1, X2]= T(k,2). Further-more, Autkk[X1, X2] is the amalgamated free product of Aff(k,2) and J(k,2).The Jung-van der Kulk theorem was first proved for fields of characteristic zero by Jung in 1942. In 1953 van der Kulk extended Jung's result to fields of arbitrary characteristic. From 1960s to 1990s, people gave many different proofs of the classical theorem. The methods given by Rentschler and Abhyankar-Moh are most typical. In Section 3.1, we introduce Jung-van der Kulk theorem and the proofs given by Rentschler and Abhyankar-Moh are showed in detail. Rentschler gave a theorem of classification of local nilpotent derivations of the polynomial algebra in dimension 2.Theorem 3.2 (Rentschler) Let 0≠D be a locally nilpotent derivation of k[X1,X2]. Then there exist h∈T(k,2) and f(X2)∈k[X2] such that h-1Dh= f(X2)(?)X1.This theorem is an important result in derivation theory of polynomial algebras, from which Rentschler reproved the Jung-van der Kulk theorem. On the other hand, Abhyankar and Moh proved that every embedding of a line in affine space of dimension 2 is rectifiable, the result can be described in algebraic language as follows:Theorem 3.3 (Abhyankar, Moh) Let f(T),g(T)∈k[T], with n= degf(T),m= degg(T), such that k[f(T), g(T)]= k[T]. Then n divides m, or m divides n.Using this result, Abhyankar and Moh gave a brief proof of the Jung-van der Kulk theorem.In the case of dimension 3, in 1972, Nagata gave an automorphismsσas follows,σ(X1)=X1-2(X1X3+X22)X2-(X1X3+X22)2X3,σ(X2)= X2+(X1X3+X22)X3,σ(X3)=X3. He conjectured thatσis not tame. In 2004, Shestakov and Umirbaev finally verified Nagata's conjecture. So the Tame Generators Problem has a negative answer in dimension 3. The Tame Generators Problem is still open in dimension n> 3. In Section 3.2, we introduce the work of Shestakov and Umirbaev, and in addition, we also introduce a theorem (given by Derksen) on the structure of the tame automorphism group:Theorem 3.4 (Derksen) Let n≥3. Then T(R,n) is generated by Aff(R,n) and the elementary automorphisms E1(cX22)=(X1+cX22,X2,...,Xn), with c∈R. Furthermore, if R has sufficiently many units, then T(R, n)=< Aff(R, n),ε>, whereε= (X1+X22, X2,..., Xn).It is usually hard to determine if an automorphism is tame. Some other properties of tame automorphisms were also studied. For example, people proposed the so-called "Multi-degree Problem of Tame Automorphisms":Let F= (F1,..., Fn) be a polynomial map and mdegF:=(degF1,...,degFn) the multidegree of F. Then for which sequence (d1,...,dn) there exists a tame automorphism F:kn→Kn such that mdegF=(d1,...,dn).In dimension 2, the answer to the Multidegree Problem of Tame Automorphisms fol-lows from the Jung-van der Kulk theorem:the sequence of positive integers(d1,d2) is a multidegree of some tame automorphism if and only if d1│d2 or d2│d1. Karas studied the problem for dimension 3, and in 2009, he got the following results:Theorem 4.1 Denote by mdeg(T(k,3)) the set of multidegrees of tame automorphisms on k3. If 3≤d2≤d3, then (3, d2, d3)∈mdeg(T(k,3)) if and only if 3│d2 or d3∈3N+d2N.Theorem 4.2 Let d3≥p2≥P1≥3. If p1,p2 are prime numbers, then(p1,p2,d3)∈mdeg(T(k,3)) if and only if d3∈p1N+p2N.Theorem 4.3 (a). If d3≥7, then (5,7, d3)(?) mdeg(T(k,3)) if and only if d3≠8,9,11,13,16,18,21,23. (b) If d3≥11, then (5,11, d3) (?) mdeg(T(k,3)) if and only if d3≠12,13,14,17,18,19,23,24,28,29,34,39. (c) If d3≥13, then (5,13, d3) (?) mdeg(T(k,3)) if and only if d3≠14,16,17,19,21,22,24,27,29,32,34,37,42,47. (d) If d3≥11, then (7,11, d3) (?) mteg{T{k,3)) if and only if d3≠12,13,15,16,17,19,20,23,24,26,27,30,31,34,37,38,41,45,48,52,59.In Section 4.1, we introduce the work of Karas.In Section 4.2, we answer the Multide-gree Problem of Tame Automorphisms in some cases, and we prove the following results: Theorem 4.4 Let d3＞d2＞4 be integers. if d2, d3 are odd, then (4, d2, d3)∈mdeg(T(k,3)) if and only if 4| d2 or d3∈4N+d2N.Theorem 4.5 Let d3＞d2＞p1≥3 be integers. if p1 is prime and d2 is odd, then (p1, d2, d3)∈mdeg(T(k,3)) if and only if p1│d2 or d3∈p1N+d2N.