Polynomial Retracts and the Jacobian Conjecture

Details Polynomial Retracts and the Jacobian Conjecture Polynomial Retracts and the Jacobian Conjecture

1997-01-14

arxiv.org/abs/math/9701210v1

Mathematics

Commutative Algebra

Scientific paper

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}) $\varphi: K[x, y] \to K[x, y]$ such that $\varphi(K[x, y]) = R$. 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 $\varphi$ of $K[x,y]$ has invertible Jacobian matrix {\it and } fixes a non-constant polynomial, then $\varphi$ is an automorphism.

Affiliated with

Also associated with

No associations

Rating

Polynomial Retracts and the Jacobian Conjecture does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Polynomial Retracts and the Jacobian Conjecture, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Polynomial Retracts and the Jacobian Conjecture will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-621191