Reconstructing birational maps from their face functions

We first prove that any birational map, from an affine space of dimension ≥ 2 to itself, is not determined by its face functions. On the other hand, we prove that a birational map with irreducibly polynomial inverse is completely determined, within the class of all birational maps with irreducibly polynomial inverses, by its face functions. We show also how to effectively reconstruct such a map from its face functions. Supported partly by the Centre Interuniversitaireen Calcul Mathématique Algébrique.