Generalized D-nilpotent Matrices And Generalized Nearly S-embedded Subgroups

Affine algebraic geometry is a branch of algebraic geometry,and its main research objects are affine space and polynomial maps on it.The Jacobian conjecture and the Tame generator problem are two well-known open problems in the field of affine algebraic geometry.Polynomial automorphisms are important tools for studying affine algebraic geometry.On the other hand,polynomial automorphisms and the structure of polyno-mial automorphism group are also important research topics.The topics in this article originate from the study of polynomial automorphisms.Let K be a field of characteristic 0,K[X]is an n-ary polynomial ring,F:Kn?Kn a polynomial map.If F is an invertible map and its inverse is still a polynomial map,then F is called an invertible polynomial map or a polynomial automorphism.Denote by JF the Jacobian matrix of F.The Jacobian conjecture asserts that if det JF ?K\{0},then F is an invertible polynomial map.The earliest form is a question formulated by O.-H.Keller in 1939.The Jacobian conjecture has attracted the attention of several celebrated mathematicians and has been studied extensively.However,it is still open when n?2.At the end of the 20th century,Fields Medal winner Smale listed the Jacobian conjecture as one of the 18 open mathematical problems in the 21st century.It is sufficient to consider cubic linear mapping F=X+(AX)*3 to attack the conjecture,where A is an n by n matrix such that JF is nilpotent.Describing and constructing a matrix satisfying the above conditions is of great significance to the study of the Jacobian conjecture.Let VA={u?Kn|(diag(u)A)n=0}.Gorni et al.introduced and described D nilpotent matrices(i.e.,dim VA=n),Tian Yan introduced and described quasi-D nilpo-tent matrices(i.e.,VA contains an n-1 dimentional linear subspace),and Li Yueyue introduced and studied qd-nilpotent matrices(i.e.,VA is a quadratic hypersurface).To develop further research following this way,in the second chapter of this dissertation we introduce and study 2qd nilpotent matrices,that is,VA contains an n-2 dimensional linear subspace.Of course,other motivation for studying 2qd nilpotent matrices come from linear triangulation problem of quadratic linear automorphism.We first introduce 2qd nilpotent matrices to generalize quasi-D nilpotent matrices.It is proved that an n by n matrix with a quasi-D nilpotent n-1 by n-1 principal submatrix is 2qd nilpotent,and 2qd nilpotent matrices that are not of the form mentioned are not invertible.Then we give a basic property of the Frobenius canonical form of a 2qd nilpotent matrix.It is also proved that a 3 by 3 2qd nilpotent matrix is exactly a matrix with a zero principal minor.The case of 4 by 4 2qd nilpotent matrices is rather complicated,partial results on which are in the appendix.Finally,we give the relationship satisfied by the principal minors of a completely 2qd nilpotent matrix.Two-dimensional polynomial automorphisms are all tame(Jung-van der Kulk theo-rem).For the case of dimension greater than 2,are all polynomial automorphisms tame?which is called the“Tame generator problem”.Over a field of characteristic 0,Shestakov and Umirbaev solved the three-dimensional tame generator problem negatively by af-firming the Nagata's conjecture.This work is deemed to be a significant breakthrough in the field of affine algebraic geometry.However,tame generator problems for the case of dimension greater than 3 are still open.It is well known that linearly triangulable polyno-mial automorphisms are tame.Since tame automorphism is very complicated,studying linearly triangulable automorphisms is an important way to understand tame automor-phisms.However,even when the co-rank of A is 2,it is unknown whether a quadratic linear automorphism F=X+(AX)*2 is linearly triangulable or not.We found that such a matrix A is 2qd nilpotent,which becomes the other motivation for us to study 2qd nilpotent matrices.In addition,since 2011,Karas and others used Shestakov and Umirbaev's theory to study when an increasing sequence of positive integers(d1,d2,d3)is the multiple degree of a tame automorphism,and many interesting results have been obtained.In the third chapter of this dissertation we consider the case where d1 or d2 is odd,and necessary and sufficient conditions for(d1,d2,d3)to be a multiple degree of a tame automorphism under certain restrictions are given,which generalizes some results in the literature.The structure of polynomial automorphism group is quite complicated.We know that the n dimensional general linear group is a subgroup of the n dimensional polyno-mial automorphism group.A natural idea is to examine special subgroups of polynomial automorphism groups from the viewpoint of general group theory.Chapter 4 of this dis-sertation is such an attempt.Synthesizing concepts of nearly M-supplemented subgroup and nearly S-embedded subgroup,we introduce the following new embedding propery of subgroups in large groups,that is,generalized nearly S-embedding property of subgroups.Let G be a finite group,and H?G.If there exist K,T?G such that T and HT are S-permutable in G,H ?T ?K?(H)?H and K is S-semipermutable in G,then H is called a generalized nearly S-embedded subgroup of G.Using generalized nearly S-embedded subgroups we give sufficient conditions for a group to be a p-supersolvable group or a supersolvable group.Then all p-chief factors of some finite groups are determined.Finally,some corollaries from the main results of this chapter are listed,which shows that main results of this chapter generalize many results in the literature.