# Dynamical method in algebra: Effective Nullstellens\"atze

## TL;DR

This paper introduces a general dynamical approach to derive effective Nullstellensatz and Positivstellensatz results, providing constructive proofs and algebraic identities in algebraically closed valued fields and ordered groups.

## Contribution

It develops a unified dynamical method for producing effective algebraic certificates and constructive proofs of classical algebraic results previously reliant on non-constructive principles.

## Key findings

- New effective Nullstellensatz and Positivstellensatz results in valued fields and ordered groups.
- Constructive versions of classical algebraic theorems like total ordering of real fields.
- Development of the concepts of dynamical proofs and simultaneous collapse.

## Abstract

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.