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

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

2004-04-14

arxiv.org/abs/math/0404268v2

Mathematics

Number Theory

25 pages, minor corrections from version 1, to appear in International Journal of Number Theory

Scientific paper

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.

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