Global injectivity of maps and related topics

• Grant number: 20/14498-4
• Support Opportunities: Regular Research Grants
• Start date: April 01, 2021
• End date: March 31, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Francisco Braun
• Grantee: Francisco Braun
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil
• Associated researchers: Bruna Orefice Okamoto ; Jaume Llibre Salo ; Jean Venato Santos ; José Ruidival Soares dos Santos Filho ; Luis Renato Gonçalves Dias

Abstract

Let F: R^n --> R^n be a local diffeomorphism. This project contains some directions toward determining additional conditions guaranteeing the global injectivity of F. The mechanisms of investigation are inside or are related to the area of dynamical systems, mainly concerning the qualitative theory of differential equations. We present three lines of research related to our projects along the last years with Brazilian and foreign collaborators. The first one deals with the polynomial case in R^2 and reach conditions related to the degree of F as well as the study of foliations of R^2 rising from polynomial submersions. The second one still deals with the polynomial case in R^2, but now under the viewpoint of the relations that the problem has with centers of polynomial Hamiltonian vector fields. Finally, the third line of research deals with the n-dimensional case, with n greater than or equal to 3, as well as with some aspects of the injectivity problem in the complex case, i.e., when in place of R^n we put C^n. (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

BRAUN, FRANCISCO; FERNANDES, FILIPE. On Reeb components of nonsingular polynomial differential systems on the real plane. Journal of Differential Equations, v. 320, p. 10-pg., 2022-05-25. (19/07316-0, 20/14498-4)

BRAUN, FRANCISCO; FERNANDES, FILIPE. Very degenerate polynomial submersions and counterexamples to the real Jacobian conjecture. Journal of Pure and Applied Algebra, v. 227, n. 8, p. 10-pg., 2023-02-17. (20/14498-4, 19/07316-0)

BRAUN, FRANCISCO; MEZA-SARMIENTO, INGRID S.. Topological equivalence of submersion functions and topological equivalence of their foliations on the plane: The linear-like case. Journal of Mathematical Analysis and Applications, v. 521, n. 2, p. 25-pg., 2022-12-20. (20/14498-4, 19/07316-0)