Jie-Tai Yu

Combinatorial methods : free groups, polynomials, and free algebras

Preface.- Introduction.- I. Groups: Introduction. Classical Techniques. Test Elements. Other Special Elements. Automorphic Orbits.- II. Polynomial Algebras: Introduction. The Jacobian Conjecture. The Cancellation Conjecture. Nagatas Problem. The Embedding Problem. Coordinate Polynomials. Test Polynomials.- III. Free Nielsen-Schreier Algebras: Introduction. Schreier Varieties of Algebras. Rank Theorems and Primitive Elements. Generalized Primitive Elements. Free Leibniz Algebras.- References.- Notations.- Author Index.- Subject Index.

Polynomial Retracts and the Jacobian Conjecture

VLADIMIR SHPILRAIN AND JIE-TAI YUAbstract. Let K[x,y] be the polynomial algebra in two variables over a fieldK of characteristic 0. A subalgebra R of K[x,y] is called a retract if there is anidempotent homomorphism (a retraction, or projection) ϕ : K[x,y] → K[x,y] suchthat ϕ(K[x,y]) = R. The presence of other, equivalent, definitions of retractsprovides several different methods of studying them, and brings together ideasfrom combinatorial algebra, homological algebra, and algebraic geometry. In thispaper, we characterize all the retracts of K[x,y] up to an automorphism, andgive several applications of this characterization, in particular, to the well-knownJacobian conjecture. Notably, we prove that if a polynomial mapping ϕ of K[x,y]has invertible Jacobian matrix and fixes a non-constant polynomial, then ϕ is anautomorphism.

Degree estimate for subalgebras generated by two elements

We develop a new combinatorial method to deal with a degree estimate for subalgebras generated by two elements in different environments. We obtain a lower bound for the degree of the elements in two-generated subalgebras of a free associative algebra over a field of zero characteristic. We also reproduce a somewhat refined degree estimate of Shestakov and Umirbaev for the polynomial algebra, that plays an essential role in the recent celebrated solution of the Nagata conjecture and the strong Nagata conjecture.

Test elements and retracts of free Lie algebras

An element it of a finitely generated free Lie algebra L is a test element if any endomorphism of L fixing u is an automorphism. We prove that test elements of L are precisely those elements not contained in any proper retract of L. In addition we characterize retracts of free Lie algebras.

The Embedding Problem

Let K[x1, ... , x n ] be the polynomial algebra in n variables over a field K. Any collection of polynomials p1, ... , p m from K[x1, ... , x n ] determines an algebraic variety {p i = 0, i = 1, ... , m} in the affine space K n . We shall denote this algebraic variety by V (p1, ... , p m ).

Generic, almost primitive and test elements of free Lie algebras

We construct a series of generic elements of free Lie algebras. New almost primitive and test elements were found. We present an example of an almost primitive element which is not generic.

D-resultant for rational functions

In this paper we introduce the D-resultant of two rational functions f(t), g(t) ∈ K(t) and show how it can be used to decide if K(f(t), g(t)) = K(t) or if K[t] C K[f(t),g(t)( and to find the singularities of the parametric algebraic curve define by X = f(t),Y = g(t). In the course of our work we extend a result about implicitization of polynomial parametric curves to the rational case, which has its own interest.

Test Elements, Retracts and Automorphic Orbits of Free Algebras

A nonzero element a of an algebra A is called a test element if for any endomorphism φ of A it follows from φ(a)=a that φ is an automorphism of the algebra A. A subalgebra B of A is a retract if there is an ideal I of A such that A=B ⊕ I. We consider the main types of free algebras with the Nielsen–Schreier property: free nonassociative algebras, free commutative and anti-commutative nonassociative algebras, free Lie algebras and superalgebras, and free Lie p-algebras and p-superalgebras. For any free algebra F of finite rank of such type we prove that an element u is a test element if and only if it does not belong to any proper retract of F. Test elements for monomorphisms of F are exactly elements that are not contained in proper free factors of F. These results give analogs of Turners results on test elements of free groups. We also characterize retracts of the algebra F. We prove that if some endomorphism φ preserve the automorphic orbit of some nonzero element of F, then φ is a monomorphism. For free Lie algebras and superalgebras over a field of characteristic zero and for free Lie p-(super)algebras over a field of prime characteristic p we show that in this situation φ is an automorphism. We discuss some related topics and formulate open problems.

The Strong Anick Conjecture is true

Recently Umirbaev has proved the long-standing Anick conjecture, that is, there exist wild automorphisms of the free associative algebra

On generators of polynomial algebras in two commuting or non-commuting variables

K\langle x,y,z\rangle