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Dynamical method in algebra: Effective Nullstellensätze

Michel Coste, Henri Lombardi, Marie-Françoise Roy

TL;DR

The paper develops a unifying dynamical approach to extract constructive content from classical algebraic proofs, applying it to Hilbert’s Nullstellensatz, Stengle’s Positivstellensatz, and new Positivstellensätze for valued fields and ordered groups. By recasting these theorems as simultaneous collapses across theories and providing explicit algebraic certificates, it yields decision procedures and constructive certificates that mirror non-constructive existence claims. The framework connects to coherent topos theory, via classifying topoi, and offers concrete algorithmic tools (e.g., leafwise trees, Hörmander tableaux) to decide emptiness and produce explicit identities. The results enable constructive embeddings, real-closure type certainties, and integral-closure-type conclusions, thereby expanding the scope of constructive algebra in algebraic geometry, valuation theory, and ordered-group settings with practical certificates. This synthesis advances a program of constructive reinterpretation of classical theorems using dynamical proofs and explicit algebraic certificates.

Abstract

We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups. These various effective Nullstellensätze 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.

Dynamical method in algebra: Effective Nullstellensätze

TL;DR

The paper develops a unifying dynamical approach to extract constructive content from classical algebraic proofs, applying it to Hilbert’s Nullstellensatz, Stengle’s Positivstellensatz, and new Positivstellensätze for valued fields and ordered groups. By recasting these theorems as simultaneous collapses across theories and providing explicit algebraic certificates, it yields decision procedures and constructive certificates that mirror non-constructive existence claims. The framework connects to coherent topos theory, via classifying topoi, and offers concrete algorithmic tools (e.g., leafwise trees, Hörmander tableaux) to decide emptiness and produce explicit identities. The results enable constructive embeddings, real-closure type certainties, and integral-closure-type conclusions, thereby expanding the scope of constructive algebra in algebraic geometry, valuation theory, and ordered-group settings with practical certificates. This synthesis advances a program of constructive reinterpretation of classical theorems using dynamical proofs and explicit algebraic certificates.

Abstract

We give a general method for producing various effective Null and Positivstellensätze, and getting new Positivstellensätze in algebraically closed valued fields and ordered groups. These various effective Nullstellensätze 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.

Paper Structure

This paper contains 19 sections, 50 theorems, 73 equations.

Table of Contents

  1. Dynamical proofs
  2. Dynamical theories and dynamical proofs
  3. Collapse
  4. Dynamical theories and coherent toposes
  5. Hilbert's Nullstellensatz
  6. Direct theories
  7. Some simultaneous collapses
  8. Decision algorithm and constructive Nullstellensatz
  9. Provable facts and algebraic theory of quasi-domains
  10. Stengle's Positivstellensatz
  11. Some simultaneous collapses
  12. Decision algorithm and constructive Positiv-stellensatz
  13. Provable facts and generalized Positivstellensätze
  14. A Positivstellensatz for valued fields
  15. Some simultaneous collapses
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let ${\cal D}$ be a dynamical theory in the language ${\cal L}$, $(G;R)$ a presentation and $B({\bf{t}})$ a fact of $(G,R)$. There is a construction associating to every proof of $R \,\vdash\, B({\bf{t}})$ in the classical first order theory ${\cal D}$ a dynamical proof of $B({\bf{t}})$.

Theorems & Definitions (64)

  • Definition 1
  • Theorem 1.1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1.5
  • Proposition 2.1
  • Proposition 2.2
  • Remark 2.3
  • Theorem 2.4
  • ...and 54 more