Outer Rank, Retracts And Dependence Problem Of Polynomials

The outer rank of a polynomial f(X)∈k[X] is the minimum number of variableson which automorphic images of f can depend. It is an important research topic tospecifically tell the outer rank of an element of an algebraic system, for example, free Liealgebras and free groups. However to our knowledge, little work on outer rank is donefor polynomial rings. In this dissertation by using retracts we give the outer rank of aclass of polynomials specifically. We say an endomorphism of a polynomial ring topreserve outer rank if sends each polynomial to one with the same outer rank. By thedefinition we know that the outer rank of an element is invariant under automorphisms.Conversely, it is natural to ask whether an endomorphism preserving outer rank is anautomorphism. This problem is called outer rank preserving problem. In Chapter2ofthis dissertation we determine the outer rank of a class of polynomials firstly, and thenpartially solve the outer rank preserving problem.Theorem2.2.1Let g(x, y)=x+yq(x, y). Then Orank(g(x, y))=1if and only ifq(x, y)∈k[y]. Equivalently, Orank(g(x, y))=2if and only if q(x, y)∈k[x, y]\k[y].Theorem2.2.2Let∈End(k[x, y]). Suppose that preserves outer rank. If thereis a coordinate in (k[x, y]), then is an automorphism.Corollary2.2.4Let∈End(k[x, y]). If preserves outer rank and (u(x, y))=u(x, y) for some u(x, y)∈k[x, y]\k, then is an automorphism. Suppose that Ris a subalgebrg of k[X].We call R is a retract of k[X]if R is the image of an idempotent of End(k[X]).Retract is a subject with great theoretical interests and valuable applications.In1977,Costa showed the following results:1)If trdegk(R)=0,then we have R=k；2)If trdegk(R)=1,then there exists g(X)∈k[X]such that R=k[g]；3)If trdegk(R)=n,then we must have R=k[X]. In particular,retracts of k[x,y]are all polynomial rings over k. Furthermore,Costa formulated the following problem:Are all the retracts of k[X]polynomial rings over k? He showed that the positive answer to the problem will imply Zariski Cancelation Conjecture,a famous unproven conjecture in algebraic geometry. In1999,by using retracts Shpilrain and Yu proved that the2-dimensional Jacobian conjecture is equivalent to the statement that k[f1]or k[f2]is a retract of k[x,y]for any2-dimensional polynomial map F=(f1(x,y),f2(x,y))satisfying the Jacobian condition. In2008,by making use of the retracts Gong and Yu solved the2-dimensional automorphic orbit problem for polynomial rings. In Chapter3of this dissertation,for the case n=3,we give partial answers to Costa's problem.Theorem3.2.1Let R be a retract of k[x,y,z]with the idempotent endomorphism φ. If trdegkR=2and one of the components of φ is a coordinate p∈{x,y,z),then there is a coordinate f and a polynomial g∈k[x,y,z],such that R=k[f,g].Let R be a subalgebrg of S. We call R a inert subalgebrg of S if for any nonzero a,b∈S with ab∈R we must have a∈R and b∈R.Theorem3.2.2Let R be a retrAct of k[x,y,z]with trdegk R=2and let R contain no coordinates.If R is an inert subalgebrg of k[x,y,z]and the corresponding retraction φ=(f,g,h)is the gradient of a polynomial,then.f,g,h are k-linearly dependent. In particular,R=k[p,q],where p,q∈{f,g,h}.Let R be a subalgebrg of S.We call R is a2-valuation algebrg if F(a,b)=0,implies a∈R or b∈R for any a,b∈S,whereTheorem3.2.3Let R be a retract of k[x,y,z]with trdegkR=2. If R is a2-valuation algebrg,then there exist p,g∈R such that R=k[p,q].Furthermore,there is an r∈k[x,y,z]such that k[x,y,z]=k[p,g,r].The Jacobian conjecture asserts that for a polynomial map F over a field k of charac-teristic0,if the Jacobian of F is a nonzero constant,then F is invertible.It attracts lots of people's attention since its formulation by Keller in1939, and now it becomes one ofthe most famous problems in afne algebraic geometry. In1999, the conjecture was listedby Smale as the sixteenth problem (there are18of them) in his "mathematical problemsfor the next century". Today, Jacobian conjecture is far from solved. In1982, Bass etal. gave a classical reduction theorem for the Jacobian conjecture. They showed that itsufces to prove the conjecture for the following polynomial maps: F=X+H, where His homogeneous of degree3with H nilpotent. This reduction theorem led to the study ofthe following HDP (n, d)(the homogeneous dependence problem): if H is a homogeneouspolynomial map of degree d with H nilpotent, then are the components of H linearlydependent over k? In2004, by studying quasi-translations de Bondt gave counterex-amples to HDP (n, d). De Bondt also studied a special class of quasi-translations. Letg(X)∈k[X] with det(Hg)=0. Then there exists0=R(Y)∈k[Y]:=k[y1, y2,..., yn]such that R(g)=0. Set G=R g. Then X+G is a quasi-translation. DeBondt formulated the following problem later in his PhD thesis: are the components ofG linearly dependent over k? This special dependence problem has positive answer forn≤3, while it is still open for n≥4. In Chapter4of this dissertation we discuss thisproblem. In particular, for the case n=4we give a necessary and sufcient condition forthe components of G to be linearly dependent.Theorem4.2.1For h∈k[x1, x2, x3, x4] with det Hh=0and a relation R of h,let H=R(h). If rkHh≤2, then the components of H are linearly dependent; IfrkHh=3and H=0, then the components of R(h) are linearly dependent if and onlyif the components of g are linearly dependent, where g is a generator of the relationideal of h.Corollary4.2.2Let h∈k[x1, x2, x3, x4] and homogeneous. If det Hh=0, then forany relation R of h, the components of R(h) are linearly dependent.Let h∈k[x1, x2,..., xn]. we call h degenerate if there exists T∈GLn(k) such thath(T X)∈k[x1, x2,..., xn1]. For the case n=5, we have the following partial result.Theorem4.2.2Let h∈k[x1, x2, x3, x4, x5] be homogeneous. If h is degenerate,then for any relation R of h, the components of R(h) are linearly dependent.