DEFINABLE HENSELIAN VALUATION RINGS

We give model theoretic criteria for the existence of there exists for all and for all there exists-formulas in the ring language to define uniformly the valuation rings O of models (K, O) of an elementary theory Sigma of henselian valued fields. As one of the applications we obtain the existence of an there exists for all-formula defining uniformly the valuation rings O of valued henselian fields (K, O) whose residue class field k is finite, pseudofinite, or hilbertian. We also obtain for all there exists-formulas phi(2) and phi(4) such that phi(2) defines uniformly k[[t]] in k((t)) whenever k is finite or the function field of a real or complex curve, and phi(4) replaces phi(2) if k is any number field.