Expansions of fields by angular functions

Abstract

Abstract. The notion of an angular function has been introduced by Zilber as one possible way of connecting non-commutative geometry with two 'counterexamples' from model theory: the nonclassical Zariski curves of Hrushovski and Zilber, and Poizat's field with green points. This article discusses some questions of Zilber relating to existentially closed structures in the class of algebraically closed fields with an angular function.

Generic automorphisms and green fields

“…We already mentioned that for stable T , the existence of T A implies that T does not have the finite cover property. Thus, Theorem 5.2 implies the following result of Evans [Ev08]. Claim.…”

“…However, in the green fields of Poizat and in the bad fields, using just weak CIT one only gains good definable control of dimension and rank. In this vein, there is the result of Evans that the green fields do not have the finite cover property [Ev08]. But in order to axiomatise the generic automorphism, we also need a definable control of 'multiplicities'.…”