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Quantifier elimination for lovely pairs of strongly geometric fields

Pablo Cubides Kovacsics, Felipe Estrada, Juan Pérez, David Rincón

Abstract

Let $T$ be a complete strongly geometric theory of fields with quantifier elimination. We show that the theory of lovely pairs of $T$ has quantifier elimination in Delon's definitional expansion by predicates for linear independence and function symbols for the corresponding coordinate functions. Apart from recovering Delon's original results for pairs of algebraically closed fields and dense pairs of algebraically closed valued fields, we obtain as particular cases, quantifier elimination for theories of dense pairs of real closed and $p$-adically closed fields.

Quantifier elimination for lovely pairs of strongly geometric fields

Abstract

Let be a complete strongly geometric theory of fields with quantifier elimination. We show that the theory of lovely pairs of has quantifier elimination in Delon's definitional expansion by predicates for linear independence and function symbols for the corresponding coordinate functions. Apart from recovering Delon's original results for pairs of algebraically closed fields and dense pairs of algebraically closed valued fields, we obtain as particular cases, quantifier elimination for theories of dense pairs of real closed and -adically closed fields.
Paper Structure (5 sections, 2 theorems, 10 equations)

This paper contains 5 sections, 2 theorems, 10 equations.

Table of Contents

  1. Setting and preliminaries
  2. Lovely pairs of strongly geometric fields
  3. Delon's expansion
  4. Auxiliary results
  5. Relative quantifier elimination

Key Result

Theorem 1.2

For any cardinal $\kappa>|T|$, the $\kappa$-lovely pairs of models of $T$ exist and they are elementary pairs. Their common theory $T_P$ is complete. In addition, if $(M, P_M)$ is a $\kappa$-saturated model of $T_P$ for a cardinal $\kappa>|T|$, then $(M, P_M)$ is a $\kappa$-lovely pair.

Theorems & Definitions (6)

  • Definition 1.1
  • Theorem 1.2: beren-vassi
  • Definition 1.3
  • proof
  • Corollary 2.1
  • proof