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This follows from basic facts about polynomials, to your generalization of a polynomial map $F: \mathbb R^n \to \mathbb R^n$. Since the Jacobian of $F$ does not vanish, its zero set $Z$ (the critical points of $F$) has measure 0, since $F$ is a polynomial. Away from critical points, the inverse image is a dimension 0 manifold, i.e. a finite set. Finally, the size of finite set is bounded by the degree of $F$.