Session S14 - Global Injectivity, Jacobian Conjecture, and Related Topics

Monday, July 12, 14:00 ~ 14:50 UTC-3

Real Jacobian mates

Janusz Gwoździewicz

Pedagogical University of Cracow, Poland   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $p$ be a real polynomial in two variables. We say that a polynomial~$q$ is a real Jacobian mate of $p$ if the Jacobian determinant of the mapping $(p,q):\mathbb{R}^2\to\mathbb{R}^2$ vanishes nowhere.

We will present a class of polynomials that do not have real Jacobian mates. In particular every polynomial such that its Newton polygon has a specific edge belongs to this class. Then we will apply this result in order to simplify the proofs in two articles cited below:

[1] F. Braun, J.R. dos Santos Filho, The real jacobian conjecture on $\mathbb{R}^2$ is true when one of the components has degree $3$, Discrete Contin. Dyn. Syst. 26, (2010) 75--87.

[2] F. Braun, B. Oré-Okamoto, On polynomial submersions of degree $4$ and the real Jacobian conjecture in $\mathbb{R}^2$, J. Math. Anal. Appl 443, (2016) 688--706.

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