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

Abstract

Let Γ\Gamma be a finitely generated subgroup of the multiplicative group \G_m^2(\bar{Q}). Let p(X,Y),q(X,Y)\in\bat{Q} be two coprime polynomials not both vanishing at (0,0)(0,0); let ϵ>0\epsilon>0. We prove that, for all (u,v)Γ(u,v)\in\Gamma outside a proper Zariski closed subset of Gm2G_m^2, the height of p(u,v)/q(u,v)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))h(p(u,v)/q(u,v))>h(1:p(u,v):q(u,v))-\epsilon \max(h(uu),h(v)). As a consequence, we deduce upper bounds for (a generalized notion of) the g.c.d. of u1,v1u-1,v-1 for u,vu,v running over Γ\Gamma.Comment: Plain TeX 18 pages. Version 2; minor changes. To appear on Monatshefte fuer Mathemati

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