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

Abstract

It is shown that a polynomial map $(P,Q)\in \mathbb{Q}[x,y{]}^2$ with $P_x{Q}_y-P_y{Q}_x\equiv 1$ 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 $d$.

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• Injectivity on one line