Quantifier elimination for lovely pairs of strongly geometric fields

TL;DR

This paper proves that the theory of lovely pairs of strongly geometric fields admits quantifier elimination in an expanded language, extending known results to real closed and p-adically closed fields.

Contribution

It establishes quantifier elimination for lovely pairs of strongly geometric fields in an expanded language, generalizing previous results for algebraically closed and valued fields.

Findings

Quantifier elimination in Delon's expanded language for lovely pairs.

Extension of results to real closed and p-adically closed fields.

Unification of various classes of fields under a common theoretical framework.

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.