  We give a general method for producing various effective Null and
Positivstellens\"atze, and getting new Positivstellens\"atze in algebraically
closed valued fields and ordered groups. These various effective
Nullstellens\"atze produce algebraic identities certifying that some geometric
conditions cannot be simultaneously satisfied. We produce also constructive
versions of abstract classical results of algebra based on Zorn's lemma in
several cases where such constructive version did not exist. For example, the
fact that a real field can be totally ordered, or the fact that a field can be
embedded in an algebraically closed field. Our results are based on the
concepts we develop of dynamical proofs and simultaneous collapse.
