Quantifier elimination on some pseudo-algebraically closed valued fields

TL;DR

This paper investigates the conditions under which certain pseudo-algebraically closed valued fields admit quantifier elimination when expanded with specific language symbols, highlighting both positive results and obstructions.

Contribution

It demonstrates quantifier elimination for some theories of pseudo-algebraically closed valued fields with expanded language, and identifies Galois-theoretic obstructions preventing it in others.

Findings

Quantifier elimination holds for some theories with expanded language.

Galois obstructions prevent quantifier elimination in certain cases.

Language expansion includes splitting coefficients, relative p-coordinate functions, and valuation division predicate.

Abstract

Adjoining to the language of rings the function symbols for splitting coefficients, the function symbols for relative $p$-coordinate functions, and the division predicate for a valuation, some theories of pseudo-algebraically closed non-trivially valued fields admit quantifier elimination. It is also shown that in the same language the theory of pseudo-algebraically closed non-trivially valued fields of a given exponent of imperfection does not admit quantifier elimination, due to Galois theoretic obstructions.