Quantifier elimination for Henselian fields relative to additive and multiplicative congruences (Q1320036)

Let \(F_ 1\) and \(F_ 2\) be valued fields with a common subfield \(F_ 0\). The author investigates conditions under which \(F_ 1\) and \(F_ 2\) are elementarily equivalent over \(F_ 0\). Such conditions are useful in the study of quantifier elimination. It is well known that it is not enough to require that the value-groups as well as the residue-fields of \(F_ 1\) and \(F_ 2\) are elementarily equivalent over those of \(F_ 0\). Generalizing earlier work of Serban Basarab, the author introduces structures of additive and multiplicative congruences and a relation between them. He calls them amc-structures for short. It is shown that various theories of Henselian fields admit elimination of quantifiers relative to amc-structures. These theories, however, are defined via rather complicated conditions.