多项式代数及相关代数的消去性质与自同构和代数K理论

本项目将利用Makar-Limanov不变量、Derksen不变量、判别式，特别是代数K理论等工具研究多项式代数及相关代数的消去性质及自同构的结构。（1）研究一些特殊多项式导子的常数环、多项式收缩、threefold代数的代数K理论，以此来考察这些代数与多项式代数的相似性，进而解决仿射几何领域中的一些相关问题。（2）利用Makar-Limanov不变量、Derksen不变量、判别式、代数K理论，确定一些有仿射几何理论背景的特殊代数及其非交换类比的消去性质。（3）研究一些指数自同构的tame性及某些非交换代数的自同构的tame性，进而确定某些代数的自同构群。

This project is concerned with cancellativity and structure of automorphisms for polynomial algebras and related algebras by means of Makar-Limanov invariant, Derksen invariant, discriminant, especially algebraic K-theory. (1) Developing algebraic K-theory of constant rings of special derivations and retracts of polynomial algebras and threefold algebras, we will examine how these rings are similar to the polynomial algebra and solve related problems in the field of affine geometry. (2) Determine cancellativity for some algebras in the field of affine geometry and their noncommutative analogues by means of Makar-Limanov invariant, Derksen invariant, discriminant, and algebraic K-theory. (3) Studying tameness of exponential automorphisms and tameness of automorphisms of noncommutative algebras, we will determine automorphism groups for some algebras.