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Universal-existential theories of fields

Sylvy Anscombe, Arno Fehm

TL;DR

The work develops a comprehensive framework for universal-existential fragments $\mathrm{Th}_{\forall\exists}$, $\mathrm{Th}_{\forall_n\exists}$, and related variants in the setting of function fields and Laurent series, providing general model-theoretic criteria and a toolkit of reductions that translate existential information into universal-existential consequences. Central techniques include a functorial notion of fragments, prenex normalization relative to fragments, and novel $p$-power coding that yields uniform translations $\tau_{\mathfrak{L}}$ between fragments, enabling transfer and decidability analyses. Applying these methods, the authors establish transfer principles from base-field existential theories to function fields $k(X)$ and to rational function fields $k(t)$, deriving both decidability implications and strong undecidability results in positive characteristic (notably for genus $\ge 2$ curves). For Laurent series fields, the paper extends Ax–Kochen–Ershov type results to multiple fragments, showing how residue-field theories control the theories of henselian valued fields and their Laurent series, with computable reductions and explicit limitations under certain hypotheses (like $R4$). Overall, the paper furnishes systematic, computability-aware tools to compare universal-existential fragments across function fields and Laurent series, linking arithmetic geometry with logical decidability questions.

Abstract

We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

Universal-existential theories of fields

TL;DR

The work develops a comprehensive framework for universal-existential fragments , , and related variants in the setting of function fields and Laurent series, providing general model-theoretic criteria and a toolkit of reductions that translate existential information into universal-existential consequences. Central techniques include a functorial notion of fragments, prenex normalization relative to fragments, and novel -power coding that yields uniform translations between fragments, enabling transfer and decidability analyses. Applying these methods, the authors establish transfer principles from base-field existential theories to function fields and to rational function fields , deriving both decidability implications and strong undecidability results in positive characteristic (notably for genus curves). For Laurent series fields, the paper extends Ax–Kochen–Ershov type results to multiple fragments, showing how residue-field theories control the theories of henselian valued fields and their Laurent series, with computable reductions and explicit limitations under certain hypotheses (like ). Overall, the paper furnishes systematic, computability-aware tools to compare universal-existential fragments across function fields and Laurent series, linking arithmetic geometry with logical decidability questions.

Abstract

We study various universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.
Paper Structure (7 sections, 41 theorems, 43 equations, 1 figure)

This paper contains 7 sections, 41 theorems, 43 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Fragments
  3. General model theoretic criteria
  4. Coding using $p$-th powers
  5. Function fields
  6. Laurent series fields
  7. Computational aspects

Key Result

Theorem 1.1

Let $k,l$ be fields of characteristic zero. If ${\rm Th}(k)={\rm Th}(l)$, then ${\rm Th}(k(\!(t)\!),v_t,t)={\rm Th}(l(\!(t)\!),v_t,t)$, and if ${\rm Th}(k)$ is decidable, then so is ${\rm Th}(k(\!(t)\!),v_t,t)$.

Figures (1)

  • Figure 1: Many-one reductions between theories

Theorems & Definitions (103)

  • Theorem 1.1: Ax--Kochen 1966
  • Theorem 1.2: AF16ADF23
  • Theorem 1.3
  • Theorem 1.4
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • ...and 93 more