Imaginaries in perfect bounded pseudo algebraically closed fields with finitely many independent valuations

TL;DR

This paper proves weak elimination of imaginaries for certain algebraically closed fields with valuations, and full elimination when only one valuation is present, using type extension and amalgamation techniques.

Contribution

It introduces a method combining type extension and amalgamation to achieve elimination of imaginaries in valued pseudo-algebraically closed fields.

Findings

Weak elimination of imaginaries for fields with finitely many valuations

Full elimination of imaginaries for fields with a single valuation

Extension of types to invariant types and an amalgamation theorem

Abstract

In this paper, we prove weak elimination of imaginaries for perfect bounded pseudo-algebraically closed fields equipped with finitely many independent valuations. Our approach combines an extension result for types to invariant types with an amalgamation theorem. As a special case, we obtain full elimination of imaginaries when the field is equipped with a single valuation.