Test polynomials, retracts, and the Jacobian conjecture
| dc.creator | Shpilrain, Vladimir | |
| dc.creator | Yu, Jie-Tai | |
| dc.date | 2004-05-10 | |
| dc.date.accessioned | 2026-07-07T05:08:06Z | |
| dc.date.available | 2026-07-07T05:08:06Z | |
| dc.description | Let K[x,y] be the algebra of two-variable polynomials over a field K. A polynomial p=p(x, y) is called a test polynomial (for automorphisms) if, whenever ϕ(p)=p for a mapping ϕof K[x,y], this ϕmust be an automorphism. Here we show that p \in C[x,y] is a test polynomial if and only if p does not belong to any proper retract of C[x,y]. This has the following corollary that may have application to the Jacobian conjecture: if a mapping ϕof C[x,y] with invertible Jacobian matrix is ``invertible on one particular polynomial", then it is an automorphism. More formally: if there is a non-constant polynomial p and an injective mapping ψof C[x,y] such that ψ(ϕ(p)) =p, then ϕis an automorphism. | |
| dc.description | 7 pages | |
| dc.identifier | https://arxiv.org/abs/math/0405179 | |
| dc.identifier | http://arxiv.org/abs/math/0405179 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/71127 | |
| dc.subject | Algebraic Geometry | |
| dc.subject | Commutative Algebra | |
| dc.subject | 14R10, 14R15 | |
| dc.title | Test polynomials, retracts, and the Jacobian conjecture | |
| dc.type | text |