Injectivity on one line
Janusz Gwo\'zdziewicz

TL;DR
This paper proves that a polynomial map with constant non-zero Jacobian that is injective on some line in the plane must be a polynomial automorphism.
Contribution
It establishes a new criterion for polynomial automorphisms based on injectivity on a line, extending understanding of the Jacobian conjecture in two dimensions.
Findings
If H|_l is injective on a line l, then H is an automorphism.
The result applies over algebraically closed fields of characteristic zero.
Provides a new condition for polynomial automorphisms in the plane.
Abstract
Let be an algebraically closed field of characteristic zero. Let be a polynomial mapping such that the Jacobian is a non-zero constant. In this note we prove, that if there is a line such that is an injection, then is a polynomial automorphism.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Algebraic Geometry and Number Theory · Advanced Algebra and Geometry
