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A weight-homogenous condition to the real Jacobian conjecture in $ {\mathbb {R}}^{2}$
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society / Volume 64 / Issue 4 / November 2021
- Published online by Cambridge University Press:
- 19 November 2021, pp. 1028-1036
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- Article
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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.
