The Definable-Absolute-Valuational-Closure Property

Source abstract

A valued field is said to have the definable-absolute-valuational-closure property if, in the language of valued fields, every definable set in the first-order structure of said valued field is dense, with respect to an induced valuation, inside a definable set in the first-order structure of the algebraic closure of said field, both with parameters in the smaller field. Every omega-free pseudo-algebraically closed non-trivially valued field with a finite exponent of imperfection is proved to have this property.