Zilber's Trichotomy Conjecture
Conjecture: If X is a strongly minimal set,
then exactly one of the
following is true about X.
- X is trivial in the sense that algebraic closure
(on a saturated
model of the theory of X) defines a degenerate pregeometry
(for any set A ⊂ X one has acl(A) = ∪ {acl({a}) | a ∈ A})
- X is essentially a vector space. That is, possibly after adding
some constant symbols to the language of X, there is an infinite group
space G bi-interpretable with X for
which every definable subset of any Cartesian power of G is
a finite Boolean combination of cosets of definable subgroups.
- X is bi-interpretable with an algebraically closed field.