Geometric Characterizations of Existentially Closed Fields with Operators
نویسندگان
چکیده
AD-field is a field with a derivation or a difference-operator, called D. In a suitable language, the theory of D-fields has a modelcompanion, whose axioms need not distinguish the two cases in which D can fall. The geometric concepts involved in describing these axioms can be used to characterize the existentially closed fields with a derivation and a difference-operator; but the class of these structures is not first-order.
منابع مشابه
Galois stratification and ACFA
We prove a direct image theorem stating that the direct image of a Galois formula by a morphism of difference schemes is equivalent to a Galois formula modulo the theory ACFA of existentially closed difference fields. As a consequence, we obtain an effective quantifier elimination procedure and a precise algebraic-geometric description of definable sets over existentially closed difference fiel...
متن کاملFields with several Commuting Derivations
The existentially closed models of the theory of fields (of arbitrary characteristic) with a given finite number of commuting derivations can be characterized geometrically, in several ways. In each case, the existentially closed models are those models that contain points of certain differential varieties, which are determined by certain ordinary varieties. How can we tell whether a given syst...
متن کاملA Note on Existentially Closed Difference Fields with Algebraically Closed Fixed Field
We point out that the theory of difference fields with algebraically closed fixed field has no model companion. By a difference field we mean a field K equipped with an automorphism σ. It is well-known ([?]) that the class of existentially closed difference fields is an elementary class (ACFA), and moreover all completions are unstable. The fixed field (set of a ∈ K such that σ(a) = a) is respo...
متن کاملPlaces of algebraic function fields in arbitrary characteristic
We consider the Zariski space of all places of an algebraic function field F |K of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of field...
متن کاملExistentially Closed Models of The Theory of Artinian Local Rings
The class of all Artinian local rings of length at most l is ∀2-elementary, axiomatised by a finite set of axioms Artl. We show that its existentially closed models are Gorenstein, of length exactly l and their residue fields are algebraically closed, and, conversely, every existentially closed model is of this form. The theory Gorl of all Artinian local Gorenstein rings of length l with algebr...
متن کامل