  We give a characterization, in terms of the residue field, of those henselian
valuation rings and those henselian valuation ideals that are diophantine. This
characterization gives a common generalization of all the positive and negative
results on diophantine henselian valuation rings and diophantine valuation
ideals in the literature. We also treat questions of uniformity and we apply
the results to show that a given field can carry at most one diophantine
nontrivial equicharacteristic henselian valuation ring or valuation ideal.
