Dynamical method in algebra: effective Nullstellensatze

We give a general method for producing various effective Null and Positivst ellensatze, and getting new Positivstellensatze in algebraically closed val ued fields and ordered groups. These various effective Nullstellensatze pro duce algebraic identities certifying that some geometric conditions cannot be simultaneously satisfied. We produce also constructive versions of abstr act classical results of algebra based on Zorn's lemma in several cases whe re such constructive version did not exist. For example, the fact that a re al field can be totally ordered, or the fact that a field can be embedded i n an algebraically closed field. Our results are based on the concepts we d evelop of dynamical proofs and simultaneous collapse. (C) 2001 Elsevier Sci ence B.V. All rights reserved.