Retracts, Test Elements And Automorphic Orbit Problem For Polynomial Algebras

Affine algebraic geometry is an interesting subject, which mainly studies affine spaces. Affine algebraic geometry has great relationship with the study of automorphisms of poly-nomial algebras, and with the development of affine algebraic geometry, the study of auto-morphisms of polynomial algebras becomes increasingly active.In recent years, retracts, test elements and automorphic orbit problem for polynomial algebras were largely studied, and many important results were obtained. In this thesis, we give a survey of the research on retracts, test elements and the automorphic orbit problem for polynomial algebras, and we mainly focus on the relationship between retracts and the Ja-cobian Conjecture, the relationship between retracts and test polynomials and the settlement of the automorphic orbit problem for polynomial algebras in two variables. At last, we give some generalization for some results on the automorphic orbit problem.Letκbe a field of characteristic 0. The image of an idempotent homomorphism (namely,φ2=φ) ofκ[x,y] is called a retract ofκ[x,y]. In Section three we introduce the structure of retracts of polynomial algebra in two variables, and the relationship between the Jacobian Conjecture and polynomial retracts. Besides, we introduce the application of test polynomi-als to distinguishing automorphisms from non-automorphisms.Costa proved that every proper retract ofκ[x,y](i.e., one which is different from k andκ[x, y])has the form k[p] for some polynomial p∈κ[x, y]. Furthermore, Shpilrain and J.-T. Yu showed the following theorem:Theorem 3.1. Letκ[p] be a retract ofκ[x,y]. Then there exists an automorphismφofκ[x,y] that takes the polynomial p to x+yq for some polynomial q= q(x,y)∈κ[x,y]. A retraction for k[φ(p)] is given byφ:φ(x)＝x+yq;φ(y)=0.Shpilrain and J.-T.Yu proposed the following conjecture and established a link between retracts of k[x, y] and the Jacobian Conjecture:Conjecture "R" Let p,q∈k[x,y] such that the corresponding Jacobian matrix is invertible, then k[p] is a retract of k[x, y].Theorem 3.2. Conjecture "R" has an affirmative answer if and only if Jacobian Con-jecture has an affirmative answer.Let k[X]= k[x1,…, xn] be the polynomial algebra in n variables and let p∈k[X], if an endomorphismφsatisfiesφ(p)= p, then we say thatφfixes p. If every endomorphism which fixes the polynomial p is an automorphism, then p is called a test polynomial. Test polynomials have great relationship with retracts, for example, the element of k[x, y] is either a test polynomial or contained in a proper retract.In Section four we introduce the automorphic orbit problem for polynomial algebras, and one of its subproblems, namely, the coordinate preserving problem.Let An be the free associative algebra or the polynomial algebra of rank n over k, p∈An-k, andφan endomorphism of An preserving the automorphic orbit of p in An, i.e. for each automorphismα∈Aut(An), there exists an automorphismβsuch thatφ(α(p))=β(p)(all coordinates are in the same orbit).In 2008, S.-J.Gong and J.-T.Yu proposed the following problem:Problem 4.3.(Automorphic orbit problem for An) Let p∈An-k, andφan endomor-phism of An. Ifφpreserves the automorphic orbit of p, then does it follow thatφis an automorphism of An?S.-J.Gong and J.-T.Yu proved Problem 4.3 has an affirmative answer for n= 2.Theorem 4.1. If an endomorphismφof A2 preserves the automorphic orbit of a non-constant element p∈A2, thenφis an automorphism of A2.They also proved the following result.Theorem 4.2. If an element p∈A2 doesn't belong to any proper retract of A2, then p is a test element of A2. In this Section, we also introduce an subproblem of the automorphic orbit problem, namely, the coordinate preserving problem.Problem 4.4.(Coordinate preserving problem) Letφbe an endomorphism of Pn. If for each coordinate f of Pn,φ(f) is a coordinate of Pn, then isφan automorphism? This problem was proposed by Shpilrain and van den Essen, and it was related to the Jacobian Conjecture.Theorem 4.3. If the Jacobian Conjecture has an affirmative answer in dimension n - 1, then Problem 4.4 has an affirmative answer in dimension n, in case k is an algebraically closed field.In the proof of Theorem 4.3, the authors only used the condition thatφis an endo-morphism sending linear coordinates to coordinates. So people formulated the following problem:Problem 4.5. Letφbe an endomorphism of k[X] sending all linear coordinates to coordinates, does it follow thatφis an automorphism?When k is an algebraically closed field, there are two results as follows:(1) If Jacobian Conjecture has an affirmative answer for dimension n - 1, then Problem 4.5 has an affirmative answer in dimension n.(2) If n= 2, then Problem 4.5 has an affirmative answer.For n≥3 and in case k is a non-algebraically closed field there is a result as follows:Theorem 4.4. For any n≥3 and any non-algebraically closed field k, Problem 4.5 has a negative answer.At present, for n≥3 and in case k is an algebraically closed field, Problem 4.5 hasn't been settled. But when k is an algebraically field of characteristic 0, Problem 4.4(namely, coordinate preserving problem)has been settled by Jelonek.In the last subsection, we give some generalization for some results on the coordinate preserving problem. We prove the following result.Theorem 4.6. For n= 2 and any field k, Problem 4.4 has an affirmative answer.The outer rank of a polynomial p is the minimum number of generators on which an automorphic image of p can depend. Based on the coordinate preserving problem, we for-mulate the following problem.Problem 4.6. Let k be an algebraically closed field of characteristic 0, and Pn the polynomial algebra in n variables over k. Letφbe an endomorphism of Pn, ifφcan preserve the outer rank of each polynomial, namely,φtakes each polynomial with outer rank r to a polynomial with outer rank r, then isφan automorphism?Whenφsatisfies the Jacobian condition(namely, detJφ∈k*), we give an affirmative answer to Problem 4.6. In fact, we prove the following result.Theorem 4.7. Letφbe an endomorphism of pn, ifφtakes each polynomial with outer rank 1 to a polynomial with outer rank 1, and detJφ∈k*, thenφis an automorphism.