Simultaneous approximation by conjugate algebraic numbers in fields of transcendence degree one

Abstract

We present a general result of simultaneous approximation to several transcendental real, complex or p-adic numbers xi_1,...,xi_t by conjugate algebraic numbers of bounded degree over Q, provided that the given transcendental numbers xi_1,...,xi_t generate over Q a field of transcendence degree one. We provide sharper estimates for example when xi_1,...,xi_t form an arithmetic progression with non-zero algebraic difference, or a geometric progression with non-zero algebraic ratio different from a root of unity. In this case, we also obtain by duality a version of Gel'fond's transcendence criterion expressed in terms of polynomials of bounded degree taking small values at xi_1,...,xi_t.