We construct and study structures imitating the field of complex numbers with exponentiation. We give a natural, albeit non first-order, axiomatisation for the corresponding class of structures and prove that the class has a unique model in every uncountable cardinality. This gives grounds to conjecture that the unique model of cardinality continuum is isomorphic to the field of complex numbers with exponentiation.
A uniform version of the Schanuel conjecture is discussed that has some model-theoretical motivation. This conjecture is assumed, and it is proved that any 'non-obviously-contradictory' system of equations in the form of exponential sums with real exponents has a solution.
Abstract. We characterize the Zariski topologies over an algebraically closed field in terms of general dimension-theoretic properties. Some applications are given to complex manifold and to strongly minimal sets.
IntroductionThere is a class of theorems that characterize certain structures by their basic topological properties. For instance, the only locally compact connected fields are R and C. These theorems refer to the classical topology on these fields. The purpose of this paper is to describe a similar result phrased in terms of the Zariski topology.The results we offer differs from the one considered above in that we do not assume in advance that our structure is a field or that it carries an algebraic structure of any kind. The identification of the field structure is rather a part of the conclusion.Because the Zariski topologies on two varieties do not determine the Zariski topology on their product (and indeed the topology on a one-dimensional variety carries no information whatsoever), the data we require consists not only of a topology on a set X, but also of a collection of compatible topologies on X" for each n . Such an object will be referred to here as a geometry. It will be called a Zariski geometry if a dimension can be assigned to the closed sets, satisfying certain conditions described below. Any smooth algebraic variety is then a Zariski geometry, as is any compact complex manifold if the closed subsets of X" are taken to be the closed holomorphic subvarieties.If X arises from an algebraic curve, there always exist large families of closed subsets of X2 ; specifically, there exists a family of curves on X2 such that through any two points there is a curve in the family passing through both and another separating the two. An abstract Zariski geometry with this property is called very ample. By contrast, there are examples of analytic manifolds X such that X" has very few closed analytic submanifolds and is not ample. Precise definitions will be given in the next section; the complex analytic case is discussed in §4. Our main result is
We describe definable sets in the field of reals augmented by a predicate for a finite rank multiplicative group Γ of complex numbers contained in the unit circle S. This structure interprets the quotient-space S/Γ which, for Γ infinite cyclic, is related to the quantum torus. Every definable set is proved to be a Boolean combination of existentially definable sets. We give a complete set of axioms for the theory of such a structure.
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