Matrices With Few Nonzero Principal Minors And Linear Triangularization Of Polynomial Maps

Affine algebraic geometry is a branch of algebraic geometry,which focuses on affine spaces and polynomial maps over them.Recognization and structure of polynomial automorphisms are two basic topics in the field.Correspondingly,there are two famous open problems:Jacobian Conjecture and Tame generators problem.Jacobian Conjecture A polynorial map with the constant Jrcobian is invertible.Researches on Jacobian Conjecture relate several branches of mathematics,and have attracted many researchers.Until now,the Jacobian Conjecture is still open for dimension n>2,and it is reduced to the case of Druzkowski maps.To study Druzkowski maps,Gorni et al.introduced D-nilpotent matrices and proved that a D-nilpotent matrix is permutation-similar to a strictly upper triangular matrix.To get more general Druzkowski maps,we introduce and study quasi-D-nilpotent matrices,which are generalization of D-nilpotent matrices.In Chapter 2,we first characterize quasi-D-nilpotent matrices in terms of principal minors,then we determine irreducible quasi-D-nilpotent matrices,and finally we give Frobenius normal forms of quasi-D-nilpotent matrices:where Ail,A44,U are strictly upper triangular matrices,a E {0,1},u,v ? Kr with all components nonzero,and ?,??Ks.In Chapter 3,we attempt to find Druzkowski matrices from quasi-D-nilpotent matrices.How-ever,it is much more difficult than expected even in the case of irreducible quasi-D-nilpotent matrices.We give a necessary and sufficient condition for a quasi-D-nilpotent matrix to be a Druzkowski matrix,and determine several special kinds of Druzkowski matrices.We prove that amatrix A=(?)is a Druzkowski matrix if and only if either v(uT)*3 = 0,or a = 0 and?(D2U)iD2?T = 0,i ?0,1,...,n-r-1,where a ? K,u,v? KT,?,??Kn-r,U is strictly upper triangular,D2 = diag(?TvX1+ UX2)*2,X1 =(x1,x2,...,xr)T,X2 =(xr+1,xr+2,...,xn)T.Tame generators problem Are polynomial automorphisms tame?A polynomial automorphism is called tame,if it is a finite composition of affine ones and triangular ones,where a triangular automorphism is ones of the form:F =(x1 ?P1,x2 +P2,...,xn+Pn)with Pi ?K[xi+1,x2+2...,xn]for 1? i ?n-1,and Pn ? K.Jung-van der Kulk Theorem affirmatively solved Tame generators problem of dimension 2.Shestakov and Umirbaev negatively solved the case of dimension 3.The problem for higher dimen-sions is still open.A polynomial map F is called linearly triangularizable,if there exists an invertible linear map T ? GLn(K),such that T-1 o F oT is a triangular automorphism.If H is a homogeneous polynomial map,then F = X+H satisfying Jacobian condition det JF =1 is equivalent to JH being nilpotent.It leads to the study of nilpotent Jacobian matrices.Nilpotent Jacobian matrices are rather complex.Therefore,people study a subclass of nilpotent Jacobian matrices:strongly nilpotent Jacobian matrices.Essen and Hubbers established the connection between linear triangularization and strong nilpotency by proving a polynomial map F = X + H is linearly triangularizable if and only if JH is strongly nilpotent.Jietai Yu proved JH is strongly nilpotent if and only if the set of Jacobian matrices JH(K):= {JH(u)| u ? Kn} is(simultaneously)triangularizable.Triangularization of a matrix set has been extensively studied in matrix theory,which motivates our study on the strong nilpotency of Jacobian matrices from the viewpoint of triangularization of matrices.The known sufficient conditions on triangularization of matrices are essentially generalization of commutativity.Hence we consider the set of matrices satisfying a permutation identity.Given a permutation ? ? Sn,a set S of matrices is called ?-permutable if a1a2…an=a?(1)a?(2)…a?(n)for all a1,a2,...,an?S.If S is ?-permutable for a nonidentical permutation ?,then S is called per mutable.For ? E Sn,let d(?)= gcd(?(1)-1,?(2)-2,..,?(n)-n).In Chapter 4,we give conditions for a permutable set S of matrices over a filed K to be linearly triangularizable through studying the permutability group of S.We prove the following results.1.Let K be the complex number field.If S is ?-permutable for a permutation ? with d(?)=1,then S is linearly triangularizable.2.Let K be an arbitrary field,and S be a set of nilpotent matrices.Then S is triangularizable if and only if ?-permutable for a permutation ? with d(?)= 1.3.If K is a field with characteristic 0,and S is a linear subspace of nilpotent matrices.If S is permutable,then S is triangularizable.As applications,we give conditions for a polynomial map to be linearly triangularizable,which generalizes some known results.