New classes of polynomial maps satisfying the real Jacobian conjecture in ${ℝ}^2$

Abstract

Abstract: We present two new classes of polynomial maps satisfying the real Jacobian conjecture in ${ℝ}^2$. The first class is formed by the polynomials maps of the form (q(x)–p(y), q(y)+p(x)) : ${\mathbb{R}}^2⟶{\mathbb{R}}^2$ such that p and q are real polynomials satisfying p'(x)q'(x) ≠ 0. The second class is formed by polynomials maps (f, g): ${\mathbb{R}}^2⟶{\mathbb{R}}^2$ where f and g are real homogeneous polynomials of the same arbitrary degree satisfying some conditions.

Key words
injective polynomial maps; global center; real Jacobian conjecture; planar Hamiltonian systems