Generic automorphisms and green fields

Abstract

We show that the generic automorphism is axiomatisable in the green field of Poizat (once Morleyised) as well as in the bad fields which are obtained by collapsing this green field to finite Morley rank. As a corollary, we obtain "bad pseudofinite fields" in characteristic 0.In both cases, we give geometric axioms. In fact, a general framework is presented allowing this kind of axiomatisation. We deduce from various constructibility results for algebraic varieties in characteristic 0 that the green and bad fields fall into this framework. Finally, we give similar results for other theories obtained by Hrushovski amalgamation, e.g. the free fusion of two strongly minimal theories having the definable multiplicity property.We also close a gap in the construction of the bad field, showing that the codes may be chosen to be families of strongly minimal sets.

Algebro-geometric axioms for $\operatorname {DCF}_{0,m}$

“…(1) difference fields: ACFA [3] (2) differential-difference fields: DCF 0 A [2] or, more generally, DCF 0,m A [11] (3) fields in characteristic p > 0 equipped with a derivation of the n-th power of the Frobenius [6] (4) fields with commuting Hasse-Schmidt derivations in positive characteristic [7] (5) fields with free operators [14] (6) theories having a "geometric notion of genericity" [5] Acknowledgements. I would like to thank the anonymous referee for his/her helpful comments and suggestions on a previous version of this note.…”

Fields and Fusions: Hrushovski constructions and their definable groups

“…Now if n √ V is not irreducible for infinitely many n, the type of ā has to specify in which irreducible component of n √ V its green n-th root lies; this would yield 2 ℵ0 possible types. Fortunately, this does not happen [22]: For every V there is n (uniformly and definably in parameters) such that k √…”

On the model companion of partial differential fields with an automorphism

“…For many geometrically well behaved theories one can prove the existence of a model companion by adapting the axiomatization of ACF A given by Chatzidakis and Hrushovski in terms of algebro-geometric objects [7]. These so called "geometric axiomatizations" have succesfully been applied to yield the existence of model companions in several interesting theories: ordinary differential fields [29], partial differential fields (several commuting derivations) [21], [22], fields with commuting Hasse-Schmidt derivations in positive characteristic [20], fields with free operators [25], and in theories having a "geometric notion of genericity" [15].…”