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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
##### Keywords:
height; subspace theorem
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