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It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y]^2$ with$P_xQ_y-P_yQ_x \equiv 1$ has an inverse map in $\mathbb{Q}[x,y]^2$ if the fiber$P=0$ contains an infinite subset of $ d^{-1}\mathbb{Z}^2$ for an integer $d$.

Siegel's theorem on integral points and the Jacobian conjecture over the rational field