Existential rank and essential dimension of definable sets

Several natural measures of complexity can be attached to an
existentially definable ("diophantine") subset of a field. One of these
is the minimal number of existential quantifiers required to define it,
while others are of a more geometric nature. I shall define these
measures and discuss interesting interactions and behaviours, some of
which depend on properties of the field (e.g. imperfection and
ampleness). We shall see for instance that the set of n-tuples of field
elements consisting of n squares is usually definable with a single
quantifier, but not always. I will also discuss connections with
Hilbert's 10th Problem and a number of open questions.
This is joint work with Nicolas Daans and Arno Fehm.