Minimal possible counterexamples to the two-dimensional Jacobian Conjecture

• Autor: Horruitiner Mendoza, Rodrigo Manuel
• Formato: tesis de maestría
• Fecha de Publicación: 2018
• Institución: Pontificia Universidad Católica del Perú
• Repositorio: PUCP-Institucional
• Lenguaje: inglés
• OAI Identifier: oai:repositorio.pucp.edu.pe:20.500.14657/146444
• Enlace del recurso: http://hdl.handle.net/20.500.12404/14385
• Nivel de acceso: acceso abierto
• Materia: Polinomios; Geometría algebraica; Automorfismo; https://purl.org/pe-repo/ocde/ford#1.01.00

Let K be an algebraically closed field of characteristic zero. The Jacobian Conjecture (JC) in dimension two stated by Keller in [8] says that any pair of polynomials P;Q ∈ L := K[x; y] with [P;Q] := axPayQ - axQayP ∈ Kx (a Jacobian pair ) defines an automorphism of L via x-> P and y -> Q. It turns out that the Newton polygons of such a pair of polynomials are closely related, and by analyzing them, much information can be obtained on conditions that a Jacobian pair must satisfy. Specifically, if there exists a Jacobian pair that does not define an automorphism (a counterexample) then their Newton polygons have to satisfy very restrictive geometric conditions. Based mostly on the work in [1], we present an algorithm to give precise geometrical descriptions of possible counterexamples. This means that, assuming (P;Q) is a counterexample to the Jacobian Conjecture with gcd(deg(P); deg(Q)) = k, we can generate the possible shapes of the Newton Polygon of P and Q and how it transforms under certain linear automorphisms. By analyzing the minimal possible counterexamples, we sketch a path to increase the lower bound of max(deg(P); deg(Q)) to 125 for a minimal possible counterexample to the Jacobian Conjecture.