Degree Estimate and Preserving Problems

This dissertation, "Degree Estimate and Preserving Problems" by Yunchang, Li, 李云昌, was obtained from The University of Hong Kong (Pokfulam, Hong Kong) and is being sold pursuant to Creative Commons: Attribution 3.0 Hong Kong License. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation. All rights not granted by the above license are retained by the author. Abstract: Abstract of thesis entitled DEGREE ESTIMATE AND PRESERVING PROBLEMS submitted by LI Yunchang for the degree of Doctor of Philosophy at The University of Hong Kong in August 2014 When an endomorphism of an algebraic structure is an automorphism and how the structure of the automorphism group is like are two of the most attractive topics in algebra. For example, the famous Jacobian Conjecture claims that an endomorphism = (f;;f ) ofK[x;;x ], the polynomial ring inn variables 1 n 1 n with Char(K) = 0, is an automorphism if and only if det(J())2K . Relevant to this, this thesis focuses on two related topics, namely degree estimates and preserving problems on polynomial rings and free associative algebras. In Chapter 1, a general introduction is given. In Chapter 2, based on a Lemma on centralizers attributed to G.M. Bergman, an estimate of L. Makar-Limanov and J.-T. Yu on the lower bound of the degrees of the elements generated by two elements in free algebras is extended to positive characteristic case. Via this new degree estimate, it is proved that in positive char- acteristic case (1) a polynomial p2 A is a test element if and only if it does not belong to any proper retracts; (2) an endomorphism is an automorphism if it pre- serves the automorphic orbit of some non-constant polynomial and (3) the pre-image of a coordinate in A is also coordinate. These results are the extensions of some results of S.-J. Gong and J.-T. Yu. Also, the structure of Mal'tsev-Neumann alge- bra over the free groups are considered and counter-examples for two conjectures of Makar-Limanov and J.-T. Yu are given. In Chapter 3, three preserving problems are considered. For the \coordinate preserving problem," a strong version of Derksen's observation is proved that for an arbitrary eld K of characteristic zero, if an endomorphism maps each tame coordinate of K[X] to a coordinate, then has a constant jacobian. The technique is easy to understand since only Galois Theory is used. This supplies a tool tototally solve the \coordinate preserving problem." For the related \linear coordinate preserving problem," the free case when n = 2 is totally solved, and for the new direction \retracts preserving problem," the case of K[x;y] when Char(K) = 0 is considered. DOI: 10.5353/th_b5312333 Subjects: Associative algebras Polynomial rings