Approximations algébriques des points dans les espaces projectifs I Original Research Article

We establish approximation properties by algebraic points, of points in projective spaces of dimension less-than-or-equals, slant3. We introduced this kind of properties in a previous text where it was shown how they can be used to prove algebraic independence results. In dimension one, a finer approximation property has been developped by D. Roy and M. Waldschmidt then M. Laurent and D. Roy, for similar purposes. The essential tool which enables us to reach dimension three is a refined effective lower bound for the Hilbert function of a prime ideal.