Title: Counting the number of distinct distances of elements in valued field extensions,
Abstract: The defect of valued field extensions is a major obstacle in open problems in
resolution of singularities and in the model theory of valued fields, whenever
positive characteristic is involved. We continue the detailed study of defect
extensions through the tool of distances, which measure how well an element in
an immediate extension can be approximated by elements from the base field. We
show that in several situations the number of essentially distinct distances in
fixed extensions, or even just over a fixed base field, is finite, and we
compute upper bounds. We apply this to the special case of valued functions
fields over perfect base fields. This provides important information used in
forthcoming research on relative resolution problems.