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A lower bound for the height of a rational function at \(S\)-unit points. (English) Zbl 1086.11035
Let \(a,b\) be given multiplicatively independent integers and let \(\varepsilon >0\). In a recent paper Y. Bugeaud, P. Corvaja and U. Zannier [Math. Z. 243, 79–84 (2003; Zbl 1021.11001)] gave the nearly best possible upper bound \(\exp(\varepsilon n)\) for \(\gcd (a^n-1,b^n-1)\). P. Corvaja and U. Zannier [Proc. Am. Math. Soc. 131, 1705–1709 (2003; Zbl 1077.11052)] generalized this by showing that \(\gcd (u-1,v-1)<\max(| u| ,| v| )^{\varepsilon}\) with finitely many exceptions for multiplicatively independent S-units \(u,v\).
In the present paper the problem is generalized to pairs of rational functions and the estimates are formulated in terms of height functions. Let \(f(X,Y)\in \overline{Q}(X,Y)\) be a rational function and \(\Gamma\) a finitely generated subgroup in \(G_m^2(\overline{Q})\). Denote by \(T_1,\ldots, T_N\) the monomials appearing in the numerator and denominator of \(f\) and suppose that \(1\in \{T_1,\ldots, T_N\}\). Then for every \(\varepsilon>0\) the Zariski closure of the set of solutions \((u,v)\in\Gamma\) of the inequality \[ h(f(u,v))>(1-\varepsilon) \max\{ h(T_1(u,v)),\ldots, h(T_N(u,v)) \} \] is a finite union of translates of proper subtori of \(G_m^2\).

MSC:
11J25 Diophantine inequalities
11G50 Heights
11J87 Schmidt Subspace Theorem and applications
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