Keller Maps With Smoothness On One Line,Monomial Derivations And Higher Derivations

Affine algebraic geometry is a field of algebraic geometry.Polynomial maps on affine spaces are important research topics.Most of the research in this field come from several well-known open problems,such as the Jacobian Conjecture,Tame Generators Problem,Zariski's Cancellation Problem,etc.Polynomial derivations are important tools for studying polynomial maps.Polyno-mial derivations play an important role in the study of the Hilbert's 14th Problem,the Jacobian Conjecture,Zariski's Cancellation Problem.A higher derivation is an extension of derivations and have a wide range of applications in commutative algebra,ring theory,Lie algebra and algebraic geometry.A polynomial map F is called a Keller map if its Jacobian is a non-zero constant.This dissertation first proves that the invertibility of two-dimensional Keller maps is equivalent to the smoothness of its image on one line.A topological proof is also presented.It is also proved that the restriction of a Druzkowski map on a line passing through the origin is an injective map.Then we study some derivations on rings of polynomials with two or four variables.We give conditions for these derivations to have the trivial constant rings.Finally,we present a representation of a higher derivation on a ring of polynomials.This is used to give an algebraic structure of the higher derivations and to prove that components except for the first one in a higher derivation on the field of rational functions are not surjective.Kernels of higher derivations under the scalar extension are discussed.Cynk and Rusek proved that a polynomial map over an algebraically closed field is invertible if and only if it is injective.Gwozdziewicz proved that injectivity of a Keller map on C2 is equivalent to injectivity on a line.Abhyankar proved that injectivity of a Keller map on C2 is equivalent to smoothness of generic members of the pencil F(A),where A ={x = b | 6 ? C}.In Chapter 2 of this dissertation we first study a Keller map F on C2,whose image restricted on a line is smooth.It is proved that if there is a line IL in C2 such that F(L)is smooth,then F is invertible.This generalizes the result of Abhyankar.Then we prove our result in a topological method.Finally,it is proved that a Druzkowski map on Cn on a line passing through the origin is injective.In Chapter 3,we study constant rings of monomial derivations on rings of binary and quaternary polynomials.We give a necessary and sufficient condition for a binary monomial derivation to have the trivial constant ring.For a strictly ternary monomial derivation d,Nowicki proved that d has no Darboux polynomials if and only if its constant field in the field of rational functions is trivial.Under the condition of ?d?0,Nowicki generalized the result above to the case of n variables,and conjectured that the condition ?d?0 is redundant.He also pointed out that in the quaternary case,even for the derivation d(x)= t2,d(y)= zt,d(z)= y2,d(t)= xy,the answer to this problem is still unknown.However,it is easy to find that the constant ring of the derivation is not trivial.To examine the problem further,in Section 2 of this chapter we consider a wider class of derivations d(x)= Z?13t?14,d(y)= z?23t?24,d(z)= X?31y?32,d(t)= x?41y?42.We give necessary and sufficient conditions for the non-existence of binomials and trinomials in the constant ring of d.In Chapter 4,we study higher derivations over rings of polynomials and fields of rational functions.Unlike derivations,the set HS(K[X])of higher derivations over a ring of polynomials does not have a natural K[X]-module structure,but it has a noncommutative group structure,called Hasse-Schmidt group.It is known that each component of a higher derivation can be written as a K[X]-linear combination of finite products of partial derivatives.We give the exact expression of such K[X]-linear combinations.This is used to give an additive operation of higher derivations.It is proved that(HS(K[X]),?)is a commutative group,and the additive group and the Hasse-Schmidt group of HS(K[X])form a brace.Then we prove that components(except the first one)of a higher derivation on a finite extension of K(X)are not surjective.Finally,we discuss kernels of higher derivations under scalar extension,and prove that if K(?)K' is an extension of fields,D ={dm}?m=0 ? HS(K[X]),and D' = {d'm}?m=0 ? HS(K'[X]),such that d'm(xi)=dm(xi),(?)m?0,i=1,2,...,n,then tr.degK K(X)D = 0 if and only if tr.degK' K'(X)D' = 0.Furthermore,if K(?)K' is a finite extension of fields,then tr.degK Q(K[X]D)= 1 if and only if tr.degK' Q(K'[X]D')= 1.