A lower bound for the height of a rational function at S-unit points

2003/11/04 by Pietro Corvaja, Umberto Zannier, Corvaja, Pietro +1 · 1 citation
Mathematics · #11J25 #Analytic Number Theory Research #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Number Theory (math.NT) #math.NT #msc:11J25

paper · pdf · doi:10.48550/arxiv.math/0311030

Plain TeX 18 pages. Version 2; minor changes. To appear on Monatshefte fuer Mathematik

openalex publication_date 2003/11/04 · arxiv created 2004/04/22 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let Γ be a finitely generated subgroup of the multiplicative group \G_m²(Q). Let p(X,Y),q(X,Y)∈\batQ be two coprime polynomials not both vanishing at (0,0); let ε>0. We prove that, for all (u,v)∈Γ outside a proper Zariski closed subset of G_m², the height of p(u,v)/q(u,v) verifies h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-εmax(h(uu),h(v)). As a consequence, we deduce upper bounds for (a generalized notion of) the g.c.d. of u-1,v-1 for u,v running over Γ.

Citations

• Finiteness of integral values for the ratio of two linear recurrences

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• Multiplicative dependence in the sumset of multiplicative groups

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