Commutative Bases, Darboux Polynomials Of Derivations And Multidegrees Of Tame Automorphisms

Afne algebraic geometry is a branch of algebraic geometry, which mainly studiesafne spaces and polynomial maps. Polynomial derivations and automorphisms, as im-portant research tools and objects in afne algebraic geometry, have profound backgroundand have been widely used. Most of the research in afne algebraic geometry is focusedon several outstanding problems, such as the Jacobian Conjecture, the Tame GeneratorsProblem, the Cancellation Conjecture, etc. The famous Jacobian Conjecture states asfollows:The Jacobian Conjecture A polynomial map over a field of characteristic zero isinvertible if its Jacobian determinant is a nonzero constant.The statement of the Jacobian Conjecture is rather elementary. However, the meth-ods that have been employed so far for attacking this problem are rather sophisticatedand come from several diferent areas of mathematics.The Jacobian Conjecture for k[X]:=k[x₁,..., xn], the polynomial ring in n variablesover a feld k, is equivalent to the assert that1,..., nis, apart from a polynomial coor-dinate change, the only commutative k[X]-basis of Derkk[X], where Derkk[X] denotesthe set of all derivations of k[X] and idenotes the partial derivative with respect toxi. In this dissertation, we give an equivalent condition for a set of pairwise commuting derivations to be a commutative basis of Derkk[X]. Precisely, we prove that n pairwisecommuting derivations of the polynomial ring (or the power series ring) in n variablesform a commutative basis of Derkk[X] if and only if they are k-linearly independent andhave no common Darboux polynomials.Derivations and Darboux polynomials are useful algebraic methods to study poly-nomial (or rational) diferential systems. If we associate a polynomial (or rational) dif-ferential systemd the existence of Darboux polynomials of D is a necessary condition for the system tohave a frst polynomial integral, rational integral, or even the Liouville integral. Further-more, derivations without Darboux polynomials can be used to produce new examples ofnonholonomic irreducible modules over Weyl algebras, noncommutative simple rings orsimple Lie rings. However, the existence of Darboux polynomials for any derivation is,in general, a very difcult problem. In this dissertation, we deal with monomial deriva-tions of the polynomial ring in three variables, and give a criteria to determine whethera monomial derivation has Darboux polynomials or not.As the generalization of derivations, higher derivations and their kernels are closelyrelated to invariant theory and feld extension theory. Moreover, higher derivations andtheir kernels play an important role when we deal with some curves and afne surfaces.For example, a Ga-action on an afne scheme X=Spec(A) can be interpreted in termsof a locally fnite iterative higher derivation on A, and many things become easier to treatby observing the locally fnite iterative higher derivation on A associated with the Ga-action. In this dissertation, we study kernels of higher derivations under feld extension,and prove that the kernel of a higher derivation of the polynomial ring can be generatedby a set of closed polynomials.It is proved by Abhyankar that the two dimensional Jacobian conjecture is equivalentto the assert that the multidegree (d1, d2) of a polynomial map F with det JF∈k isprincipal, that is, d1|d2or d2|d1. The classical Jung-van der Kulk theorem shows thatevery automorphism of the polynomial ring in two variables is tame, and its multidegreeis principal. The research of mutlidegrees of polynomial automorphisms, especially tameautomorphisms, is of great signifcance to characterize the structure of the group of poly-nomial automorphisms of the polynomial ring. In this dissertation, we study two classesof mutidegrees of tame automorphisms of the polynomial ring in three variables. Precise- ly, we consider the following two cases:(1) The multidegree (d1, d2, d3) is an arithmeticprogression;(2) one of d1, d2, d3is a prime number.In Chapter1, we give a brief survey on topics studied in this dissertation.In Chapter2, we present some examples of derivations without Darboux polynomials,and give a criteria to determine whether a monomial derivation of k[x, y, z] has a trivialkernel. We further give an equivalent condition for monomial derivations to have noDarboux polynomials.Theorem2.3.3. Let D be an arbitrary monomial derivation of k[x, y, z]. Then thefollowing statements are equivalent.(1) D has no Darboux polynomials. This is an analogue of a well-known fact in linear algebra that a set of pairwisecommuting linear operators on a fnite dimensional vector space over an algebraicallyclosed feld have a common eigenvector. In fact, Theorem3.2.1reveals some relationsbetween commutative k[X]-bases of Derkk[X] and Darboux polynomials.Theorem3.2.2. Let D1,..., Dnbe pairwise commuting derivations of k[X]. ThenD1,..., Dnform a commutative k[X]-basis of Derkk[X] if and only if D1,..., Dnarelinearly independent over k and have no common Darboux polynomials.We point out that Theorem3.2.2is also valid for the power series ring k[[X]].In Chapter4, we study the kernels of higher derivations under feld extension.Theorem4.2.1. Let k k′be a field extension and let D={Dn}∞n=0be a higherexists a set S of closed polynomials in k[X] such that k[X]D=k[S].In Chapter5, frst, we briefy sketch Shestakov and Umirbaev's proof of the Nagataconjecture and give a survey on the research of mutlidegrees of tame automorphisms.Then we give a criteria to determine whether an arithmetic progression (a, a+d, a+2d) to be a multidegree of a tame automorphism:Theorem5.3.1. Let (a, a+d, a+2d) be an arithmetic progression of positive integers.(1) If a|2d, then (a, a+d, a+2d)∈mdeg(Tame k3). (2) If a2d, then (a, a+d, a+2d)∈/mdeg(Tame k3), except for the case of (4i,4i+ij,4i+2ij) with i, j∈N and j an odd number.We related this exceptional case to a conjecture of Drensky and Jie-tai Yu, whichconcerns with the lower bound of the degree of the Poisson bracket of two polynomials.We also consider a variation of a conjecture of Kara′s.Conjecture5.4.1. Let2<d₁≤d₂≤d₃be integers such that one of d₁, d₂, d₃is a prime number. Then (d₁, d₂, d₃)∈mdeg(Tame k³) if and only if d₁|d₂or d₃∈d₁N+d₂N.