Pencil of irreducible rational curves and Plane Jacobian conjecture

Abstract

We are concerned with the behavior of the polynomial maps $F=(P,Q)$ of ${\mathbb{C}}^2$ with finite fibres and satisfying the condition that all of the curves $aP+bQ=0$, $(a:b)\in {\mathbb{P}}^1$, are irreducible rational curves. The obtained result shows that such polynomial maps $F$ is invertible if $(0,0)$ is a regular value of $F$ or if the Jacobian condition holds.

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