arXiv subjects

Jie-Tai Yu

Publications and source records attributed to Jie-Tai Yu.

40 records · Page 3Linked to original sources

arXiv1999

Peak reduction technique in commutative algebra

The "peak reduction" method is a powerful combinatorial technique with applications in many different areas of mathematics as well as theoretical computer science. It was introduced by Whitehead, a famous topologist and group theorist, who used it to solve an important algorithmic problem concerning automorphisms of a free group. Since then, this method was used to solve numerous problems in group theory, topology, combinatorics, and probably in some other areas as well. In this paper, we give a survey of what seems to be the first applications of the peak reduction technique in commutative algebra and affine algebraic geometry.

math.AG↗

arXiv1998

Embeddings of curves in the plane

In this paper, we contribute toward a classification of two-variable polynomials by classifying (up to an automorphism of $C^2$) polynomials whose Newton polygon is either a triangle or a line segment. Our classification has several applications to the study of embeddings of algebraic curves in the plane. In particular, we show that for any $k \ge 2$, there is an irreducible curve with one place at infinity, which has at least $k$ inequivalent embeddings in $C^2$. Also, upon combining our method with a well-known theorem of Zaidenberg and Lin, we show that one can decide "almost" just by inspection whether or not a polynomial fiber is an irreducible simply connected curve.

math.AG↗

arXiv1997

Combinatorial problems about free groups and algebras

This is a survey of recent progress in several areas of combinatorial algebra. We consider combinatorial problems about free groups, polynomial algebras, free associative and Lie algebras. Our main idea is to study automorphisms and, more generally, homomorphisms of various algebraic systems by means of their action on ``very small" sets of elements, as opposed to a traditional approach of studying their action on subsystems (like subgroups, normal subgroups; subalgebras, ideals, etc.) We will show that there is a lot that can be said about a homomorphism, given its action on just a single element, if this element is ``good enough". Then, we consider somewhat bigger sets of elements, like, for example, automorphic orbits, and study a variety of interesting problems arising in that framework. One more point that we make here is that one can use similar combinatorial ideas in seemingly distant areas of algebra, like, for example, group theory and commutative algebra. In particular, we use the same language of ``elementary transformations" in different contexts and show that this approach appears to be quite fruitful for all the areas involved.

math.GR↗

arXiv1997

Polynomial Retracts and the Jacobian Conjecture

Let $ K[x, y]$ be the polynomial algebra in two variables over a field $K$ of characteristic $0$. A subalgebra $R$ of $K[x, y]$ is called a retract if there is an idempotent homomorphism (a {\it retraction}, or {\it projection}) $φ: K[x, y] \to K[x, y]$ such that $φ(K[x, y]) = R$. The presence of other, equivalent, definitions of retracts provides several different methods of studying them, and brings together ideas from combinatorial algebra, homological algebra, and algebraic geometry. In this paper, we characterize all the retracts of $ K[x, y]$ up to an automorphism, and give several applications of this characterization, in particular, to the well-known Jacobian conjecture. Notably, we prove that if a polynomial mapping $φ$ of $K[x,y]$ has invertible Jacobian matrix {\it and } fixes a non-constant polynomial, then $φ$ is an automorphism.

math.AC↗