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Analytic Nullstellensätze and the model theory of valued fields

Matthias Aschenbrenner, Ahmed Srhir

Abstract

We present a uniform framework for establishing Nullstellensätze for power series rings using quantifier elimination results for valued fields. As an application we obtain Nullstellensätze for $p$-adic power series (both formal and convergent) analogous to Rückert's complex and Risler's real Nullstellensatz, as well as a $p$-adic analytic version of Hilbert's 17th Problem. Analogous statements for restricted power series, both real and $p$-adic, are also considered.

Analytic Nullstellensätze and the model theory of valued fields

Abstract

We present a uniform framework for establishing Nullstellensätze for power series rings using quantifier elimination results for valued fields. As an application we obtain Nullstellensätze for -adic power series (both formal and convergent) analogous to Rückert's complex and Risler's real Nullstellensatz, as well as a -adic analytic version of Hilbert's 17th Problem. Analogous statements for restricted power series, both real and -adic, are also considered.
Paper Structure (32 sections, 132 theorems, 127 equations)

This paper contains 32 sections, 132 theorems, 127 equations.

Table of Contents

  1. Valuation-Theoretic Preliminaries
  2. Model-theoretic treatment of valued fields
  3. Extension of valuations
  4. Coarsening and specialization
  5. Convexity
  6. Weierstrass Systems and Infinitesimal Analytic Structures
  7. Weierstrass systems
  8. Infinitesimal $W$-struc-tures
  9. Infinitesimal $W$-struc-tures as model-theoretic structures
  10. Zero sets and vanishing ideals
  11. Germs of analytic functions
  12. A Proof of Rückert's Nullstellensatz
  13. Risler's Nullstellensatz
  14. The Theory $p\operatorname{CVF}$
  15. $p$-valued fields
  16. ...and 17 more sections

Key Result

Lemma 1.2

If $A$ is a regular local ring and $K=\operatorname{Frac}(A)$, then there is a valuation ring $\mathcal{O}$ of $K$ lying over $A$ such that the natural inclusion $A\to\mathcal{O}$ induces an isomorphism ${{\boldsymbol{k}}}\to\mathcal{O}/\smallo$.

Theorems & Definitions (214)

  • Definition 1.1
  • Example
  • Lemma 1.2
  • Lemma 1.3
  • Definition 1.4
  • Lemma 1.5
  • proof
  • Lemma 1.6
  • Lemma 1.7
  • Lemma 1.8
  • ...and 204 more