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Restricted analytic valued fields with partial exponentiation

Leonardo Ángel, Xavier Caicedo

TL;DR

We address the problem of formulating a first-order theory for non-archimedean ordered valued fields with restricted analytic functions augmented by a partial exponential on a convex subring. The authors develop $T_{an}(oxed{O- ext{exp}})$ and prove model completeness, completeness, and weak o-minimality, with quantifier elimination upon adding $ ext{log}$ and radicals; they also show every model embeds into a structure where the partial exponential extends to a total one. A key method is linking the induced exponential on the residue field to $T_{an}( ext{exp})$ and applying a Sacks-type embedding argument to treat residue-field cases, yielding a prime model and a robust prime-extension framework. Together with a final observation tying the theory to Hahn-field instances and suggesting Ax–Kochen–Ershov-type generalizations, this work provides a solid structural understanding of restricted analytic functions with partial exponentiation in non-archimedean settings.

Abstract

Non-archimedean fields with restricted analytic functions may not support a full exponential function, but they always have partial exponentials defined in convex subrings. On face of this, we study the first order theory of the class of non-archimedean ordered valued fields augmented by all restricted analytic functions and an exponential function defined in the valuation ring, which extends the restricted analytic exponential. We obtain model completeness and other desirable properties for this theory. In particular, any model embeds in a model where the partial exponential extends to a full one.

Restricted analytic valued fields with partial exponentiation

TL;DR

We address the problem of formulating a first-order theory for non-archimedean ordered valued fields with restricted analytic functions augmented by a partial exponential on a convex subring. The authors develop and prove model completeness, completeness, and weak o-minimality, with quantifier elimination upon adding and radicals; they also show every model embeds into a structure where the partial exponential extends to a total one. A key method is linking the induced exponential on the residue field to and applying a Sacks-type embedding argument to treat residue-field cases, yielding a prime model and a robust prime-extension framework. Together with a final observation tying the theory to Hahn-field instances and suggesting Ax–Kochen–Ershov-type generalizations, this work provides a solid structural understanding of restricted analytic functions with partial exponentiation in non-archimedean settings.

Abstract

Non-archimedean fields with restricted analytic functions may not support a full exponential function, but they always have partial exponentials defined in convex subrings. On face of this, we study the first order theory of the class of non-archimedean ordered valued fields augmented by all restricted analytic functions and an exponential function defined in the valuation ring, which extends the restricted analytic exponential. We obtain model completeness and other desirable properties for this theory. In particular, any model embeds in a model where the partial exponential extends to a full one.
Paper Structure (11 sections, 20 theorems, 28 equations)

This paper contains 11 sections, 20 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Valued fields, ordered fields and convex valuations
  4. The theories $T_{an}$ and $T_{an}(\exp)$
  5. The theory $T_{convex}$
  6. The theory $T_{an}(\mathcal{O}$-$\exp )$
  7. Model completeness of $T_{an}(\mathcal{O}$-$\exp )$
  8. Case $\operatorname{res}(E)=\operatorname{res}(K)$
  9. Case $\operatorname{res}(E)\not=\operatorname{res}(K)$
  10. Quantifier elimination, completeness, weak o-minimality
  11. Final observation

Key Result

Lemma 1

Let $(K,\mathcal{O})$ be an ordered valued field. Then the underlying additive group $K$ decomposes as $K=A\oplus A^{\prime }\oplus {\mathcal{O}} ,$ where $A$ is a group complement We say that $M$ is a group complement of $N$ in the field $K$ if $M$ is a subgroup of $K$ such that $K=M\oplus N$. to

Theorems & Definitions (30)

  • Lemma 1: Lexicographic decompositions of $K$
  • Lemma 2
  • Lemma 3
  • Corollary 4
  • Proposition 5
  • Lemma 6
  • Lemma 7
  • proof
  • Definition 8
  • Lemma 9
  • ...and 20 more