Nilpotency Polynomials Of Matrices And Triangularization Of Subspace Of Matrices

In this paper,we first introduce the nilpotency polynomials of matrices,and describe matrices with nilpotency quadratic polynomials.As an application,we give the necessary and sufficient conditions for these matrices to be Druzkowski matrices.Then by means of action of linear groups,we give conditions for triangularization of matrix subspaces.Finally,a constructive method is given to determine whether a non-injective endomorphism of bivariate polynomial rings has nontrivial fixed elements.The topics in this paper are derived from several open problems in the field of affine algebraic geometry:Jacobian conjecture,Tame generators problem,Zariski's cancelation problem.The Jacobian conjecture has been reduced to the case of Druzkowski mappings.To study this kind of map,Gorni et al.introduced D-nilpotent matrices,and proved that D-nilpotent matrices is permutation-similar to strictly upper triangular matrices.To get general Druzkowski mappings,we introduce nilpotency polynomials of a matrix.Let A be an n×n matrix.A polynomial f with n variables is called a nilpotency polynomial of A if(DA)n = 0 for each diagonal matrix D whose diagonal is a zero of f.We study matrices that has nilpotency polynomials,called qd-nilpotent matrices.In Chapter 3,structure and properties of nilpotency polynomials of matrices are studied.We first prove that the nilpotency polynomials of qd-nilpotent matrices that are not D-nilpotent is of degree one for each variable,and any two coprime nilpotency polynomials have no common variables.Then we characterize the qd-nilpotent matrices in terms of principal minors.We find that most of principal minors in qd-nilpotent matrices are 0.Finally,the Frobenius normal form of a qd-nilpotent matrix is presented:where A11 and A33 are strictly upper triangular and A22 is an irreducible qd-nilpotent matrix.Hence we need only consider irreducible qd-nilpotent matrix.In Chapter 4,we study matrices A with a nilpotency polynomial of degree two.It is proved that if A is irreducible,then there is a permutation matrix P such that one of the following holds.(?)PT AP = DB,where D is an invertible diagonal matrix,B is a skew symmetric matrix of rank 2 with off-diagonal entries nonzero in the leading principal of order 3.(?)PTAP =(?),where the diagonal positions are square matrices,??{0,1},rankB ? 2,B has no zero row and no zero column,U is a strictly upper triangular matrix,u,v,?,? are all row vectors,and all entries of u,v are nonzero,Finally,we give sufficient and necessary conditions for matrices of the form as in(?)to be Druzkowski matrices,and prove under certain conditions that corresponding Dru.kowski map is linearly trian-gularizable.Triangular automorphisms are essential and important polynomial automorphisms,which plays an significant role to describe the structure of polynomial automorphisms.Linear triangularization of polynomial mappings is a way to recognize tame automorphisms.Van den Essen et al.proved that if the Jacobian matrix of H is nilpotent,then the polynomial map F = X+H is linearly triangularizable if and only if the set of Jacobian matrices of H is simultaneously triangulariz-able.This motivates our studying linear triangularization of polynomial maps from viewpoint of triangularization of matrix sets.In Chapter 5,we study action of linear groups and triangularization of matrix subspaces.We prove that the subgroup generated by Q and ?n(K)contains a transvection,where Q is an invertible matrix that is not a monomial matrix and ?n(K)is the diagonal matrices with determinants 1.Then it is proved that a matrix subspace S satisfying an invertible matrix Q is nilpotent and so triangularizable,unless Q is a monomial matrix such that the determinant of its corresponding diagonal matrix is 1.In Chapter 6,we study non-injective endomorphisms and their fixed elements for the bivariate polynomial algebra and free associative algebra A2.We first give a classification of non-injective endomorphisms of A2 and then give a method to determine if non-injective endomorphisms of A2 have nontrivial fixed elements.