A weight-homogenous condition to the real Jacobian conjecture in $ {\mathbb {R}}^{2}$. Issue 4 (19th November 2021)

• Record Type: Journal Article
• Authors: Braun, Francisco; Valls, Claudia
• Is Part Of: Proceedings of the Edinburgh Mathematical Society. Volume 64:Issue 4(2021)
• Page Start: 1028
• Page End: 1036
• Publication Date: 2021-11-19
• Subjects: Real Jacobian conjecture -- global injectivity -- weight-homogeneous polynomials; 14R15 -- 35F05 -- 35A30; Mathematics -- Periodicals; 510
• DOI: 10.1017/S0013091521000766
• Languages: English
• ISSNs: 0013-0915

Abstract

Abstract: It is known that a polynomial local diffeomorphism $(f, \, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.