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Relative Quantifier Elimination for Separable-Algebraically Maximal Kaplansky Fields

Paulo Andrés Soto Moreno

TL;DR

The work proves that the common theory of equi-characteristic separable-algebraically maximal Kaplansky fields with an angular component map eliminates field quantifiers residually to the residue field and value group in a three-sorted language that includes ${\lambda}$-functions. Central to the approach is the Embedding Lemma, which enables resplendent quantifier elimination by solving embedding-extension problems, and this yields AKE-type principles for the relative theories of these fields. Consequently, equi-characteristic NIP and NIP$_n$ henselian fields with ac maps also enjoy resplendent field-quantifier elimination in the same language, and existential formulas can be reduced to the home sort. The results further provide a description of completions, stable-embeddedness of the residue/value sorts, and computability/decidability reductions to the residue field and value group theories, extending the Ax–Kochen–Ershov philosophy to this broad SAMK setting.

Abstract

Let $C$ be the class of separable-algebraically maximal equi-characteristic Kaplansky fields of a given imperfection degree, admitting an angular component map. We prove that the common theory of the class $C$ resplendently eliminates quantifiers down to the residue field and the value group, in a three sorted language of valued fields with a symbol for an angular component map and symbols for the parameterized lambda-functions. As a consequence, we obtain that equi-characteristic NIP and NIP$_n$ henselian fields with an angular component map resplendently eliminate field quantifiers in this language. We also prove that this elimination reduces existential formulas to existential formulas without quantifiers from the home sort. Finally, we draw several conclusions following the AKE philosophy for elements of the class $C$, including the usual AKE principles for $\equiv,\equiv_{\exists},\preceq,\preceq_{\exists}$ and for relative decidability.

Relative Quantifier Elimination for Separable-Algebraically Maximal Kaplansky Fields

TL;DR

The work proves that the common theory of equi-characteristic separable-algebraically maximal Kaplansky fields with an angular component map eliminates field quantifiers residually to the residue field and value group in a three-sorted language that includes -functions. Central to the approach is the Embedding Lemma, which enables resplendent quantifier elimination by solving embedding-extension problems, and this yields AKE-type principles for the relative theories of these fields. Consequently, equi-characteristic NIP and NIP henselian fields with ac maps also enjoy resplendent field-quantifier elimination in the same language, and existential formulas can be reduced to the home sort. The results further provide a description of completions, stable-embeddedness of the residue/value sorts, and computability/decidability reductions to the residue field and value group theories, extending the Ax–Kochen–Ershov philosophy to this broad SAMK setting.

Abstract

Let be the class of separable-algebraically maximal equi-characteristic Kaplansky fields of a given imperfection degree, admitting an angular component map. We prove that the common theory of the class resplendently eliminates quantifiers down to the residue field and the value group, in a three sorted language of valued fields with a symbol for an angular component map and symbols for the parameterized lambda-functions. As a consequence, we obtain that equi-characteristic NIP and NIP henselian fields with an angular component map resplendently eliminate field quantifiers in this language. We also prove that this elimination reduces existential formulas to existential formulas without quantifiers from the home sort. Finally, we draw several conclusions following the AKE philosophy for elements of the class , including the usual AKE principles for and for relative decidability.
Paper Structure (12 sections, 33 theorems, 40 equations)

This paper contains 12 sections, 33 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Separability
  4. Valuation Theory
  5. Kaplansky Theory
  6. Model Theory
  7. Existential Fragments
  8. Languages and Theories of SAMK Valued Fields
  9. Computability
  10. The Embedding Lemma
  11. Relative Quantifier Elimination
  12. Ax-Kochen-Ershov Principles

Key Result

Theorem 1.1

Let $\mathfrak{M},\mathfrak{N}$ be two models of $\texttt{SAMK}_{e}^{{\lambda},\mathop{\mathrm{ac}}\nolimits}$ and let $\mathfrak{A}$ be the $\mathscr{L}$-substructure of $\mathfrak{M}$ generated by a ${\lambda}$-closed subring $A$ of $M.$ Suppose that $\mathfrak{M}$ is $\aleph_0$-saturated and that

Theorems & Definitions (71)

  • Theorem 1.1: Embedding Lemma, cf. Theorem \ref{['emblem']}
  • Theorem 1.2: cf. Proposition \ref{['rqe']} and Corollary \ref{['formbyform']}
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.6
  • Lemma 2.7
  • proof
  • Lemma 2.8
  • ...and 61 more