The Tame Automorphism And Jacobian Conjucture

Abstract

Consider the notations for this paper as: R_x indicates a commutative ring, though in most result R_x are going to be domain, and R_x [X]□(∶=) R_x [X_1,...,X_n] the Polynomial ring in n elements overR_x. 
Here assuming that subsequent subcategories of 〖Aut〗_(R_x ) R_x [X]::Aff(R_x,n) is equal  the affine subgroup of 〖Aut〗_(R_x ) R_x [X]   including of every R_x-automorphisms F so that deg F_i = 1 ∀ i. J (R_x,n) ∶= the “de Jonquière’s” subgroup of 〖Aut〗_R R_x [X] including the R_x-automorphisms F Of the arrangement. 
F = (a_1 X_1+f_1 (X_2,...,X_n)a_2 X_2+f_2 (X_3,...,X_n) , . . . . , a_n X_n+f_n) Where one of the a_i ∈ R^*  and f_i  ∈ R_x [X_(i+1),...,X_n] for every 1 ≤i≤n-1 and f_n∈R_x.E(R_x,n)□(∶=) the subgroup of 〖Aut〗_(R_x ) R_x [X] The Elementary Automorphism generated, that is the form of the automorphisms is   F= (X_1,...,X_(i-1) X_i+a〖 ,X〗_(i+1),...,X_n, ) for some a ∈R_x [X_1,...,(X_i ) ̂,...,X_n,] and 1≤i≤ n.  
T(R ,n) □(∶=) the tame subset of 〖Aut〗_(R_x ) R_x [X]The subgroup generated by is  Aff(R_x, n) and E(R_x,n). 
We get each part of J (R_x, n) is a multiplication of a part of(R_x, n) and Elementary Automorphism finite in numbers. Therefore J(R_x, n) ⊂ T (R_x, n). Also, coupling the “de Jonquiere’s” automorphisms with appropriate one effectively verifies permutation maps that all Elementary Automorphism fit in to the subgroup of 〖Aut〗_(R_x  ) R_x [X] created by Aff (R_x, n) and J (R_x, n). Hence, having T (R_x,n) =(Aff(R_x  ,n) and J (R_x  ,n)).
In this paper here we assume the condition n = 2 and consider the R_x    a domain. In this paper proving that the T (R_x, 2) is the free merged result of Aff (R_x, 2) and J (R_x, 2) via their intersection. Moreover, we define an algorithm that determines if there is an endomorphism of polynomial of R_x [X,Y] is tame. 
By means of this process, the paper demonstrate that if R_x it is not a field, so it T(R,2) ≠〖Aut〗_(R_x  ) R_x [X,Y]. But, consider R_x  may be a field then it seems that we have impartiality, that is each in dimension two, automorphism taken over a field is tame. This is the more popular “Jung-van der Kulk theorem” (1.1.11).
 
Keywords : Automorphisms , Endomorphism of polynomial , Jacobian Conjecture.

Consider the notations for this paper as: R_x indicates a commutative ring, though in most result R_x are going to be domain, and R_x [X]□(∶=) R_x [X_1,...,X_n] the Polynomial ring in n elements overR_x.
Here assuming that subsequent subcategories of 〖Aut〗_(R_x ) R_x [X]::Aff(R_x,n) is equal the affine subgroup of 〖Aut〗_(R_x ) R_x [X] including of every R_x-automorphisms F so that deg F_i = 1 ∀ i. J (R_x,n) ∶= the “de Jonquière’s” subgroup of 〖Aut〗_R R_x [X] including the R_x-automorphisms F Of the arrangement.
F = (a_1 X_1+f_1 (X_2,...,X_n)a_2 X_2+f_2 (X_3,...,X_n) , . . . . , a_n X_n+f_n) Where one of the a_i ∈ R^* and f_i ∈ R_x [X_(i+1),...,X_n] for every 1 ≤i≤n-1 and f_n∈R_x.E(R_x,n)□(∶=) the subgroup of 〖Aut〗_(R_x ) R_x [X] The Elementary Automorphism generated, that is the form of the automorphisms is F= (X_1,...,X_(i-1) X_i+a〖 ,X〗_(i+1),...,X_n, ) for some a ∈R_x [X_1,...,(X_i ) ̂,...,X_n,] and 1≤i≤ n.
T(R ,n) □(∶=) the tame subset of 〖Aut〗_(R_x ) R_x [X]The subgroup generated by is Aff(R_x, n) and E(R_x,n).
We get each part of J (R_x, n) is a multiplication of a part of(R_x, n) and Elementary Automorphism finite in numbers. Therefore J(R_x, n) ⊂ T (R_x, n). Also, coupling the “de Jonquiere’s” automorphisms with appropriate one effectively verifies permutation maps that all Elementary Automorphism fit in to the subgroup of 〖Aut〗_(R_x ) R_x [X] created by Aff (R_x, n) and J (R_x, n). Hence, having T (R_x,n) =(Aff(R_x ,n) and J (R_x ,n)).
In this paper here we assume the condition n = 2 and consider the R_x a domain. In this paper proving that the T (R_x, 2) is the free merged result of Aff (R_x, 2) and J (R_x, 2) via their intersection. Moreover, we define an algorithm that determines if there is an endomorphism of polynomial of R_x [X,Y] is tame.
By means of this process, the paper demonstrate that if R_x it is not a field, so it T(R,2) ≠〖Aut〗_(R_x ) R_x [X,Y]. But, consider R_x may be a field then it seems that we have impartiality, that is each in dimension two, automorphism taken over a field is tame. This is the more popular “Jung-van der Kulk theorem” (1.1.11).