Dynamic computations inside the algebraic closure of a valued field

TL;DR

This paper presents algorithms for computing within the algebraic closure of a valued field, enabling effective quantifier elimination through geometric methods, and relies on the paZ.

Contribution

It introduces a novel computational approach for algebraic closure in valued fields, connecting algebraic and geometric ideas for quantifier elimination.

Findings

Algorithms for computation in algebraic closure of valued fields

Effective quantifier elimination method developed

Concrete implementation of algebraically closed valued extension

Abstract

We explain how to compute in the algebraic closure of a valued field. These computations heavily rely on the \NPAz. They are made in the same spirit as the dynamic algebraic closure of a field. They give a concrete content to the theorem saying that a valued field does have an algebraically closed valued extension. The algorithms created for that purpose can be used to perform an effective quantifier elimination for algebraically closed valued fields, which relies on a very natural geometric idea.