A small value estimate in dimension two
by
Damien Roy
Department of Mathematics, University of Ottawa

A small value estimate is a statement providing necessary conditions for the existence of a sequence of non-zero polynomials with integers coefficients taking small values at many points of an algebraic group. Such statements are desirable for applications to transcendental number theory, but only few instances of them are known at the moment. The purpose of this talk is to present a small value estimate for the product Ga x Gm of the additive group Ga by the multiplicative group Gm. We will show that if a sequence of polynomials with integer coefficients take small values at a point (xi, eta) together with its first derivatives with respect to the invariant derivation d/dx+y(d/dy), then both xi and eta are algebraic over Q. The precise statement compares favorably with constructions coming from Dirichlet's box principle.

Date received: May 16, 2008