Global Injectivity of Polynomial Maps in the plane [ Back ]

Date:
14.12.15   
Times:
15:30
Place:
UAB - Dept. Matemàtiques (C1/-128)
Speaker:
Francisco Braun
University:
Universidade Federal de São Carlos

Abstract:

Let $F= (p, q) : \R^2\longrightarrow\ R^2$ be a polynomial map such that $DF(x)$ is invertible for all $x\in \R^2$. The real Jacobian conjecture claims that under this conditions $F$ is globally injective. In 1994, S. Pinchuk gave a counterexample to the real Jacobian conjecture. In this counterexample, the polynomial $p$ has degree 10 and the polynomial $q$ has degree 25. It is then natural to ask if there exists a counterexample with lower degree. Or equivalently, what is the smallest degrees of $p$ and/or $q$ in order that the conjecture holds.

In this lecture we will explain the joint work with Bruna Oréfice-Okamoto, where we show that if the degree of $p$ is less than or equal to 4, then $F$ is injective. This work is part of a program in order to know what exactly fails in the real Jacobian conjecture.