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Zariski geometries
 Journal of the American Mathematical Society
, 1996
"... Let k be an algebraically closed field. The set of ordered ntuples from k is viewed as an ndimensional space; a subset described by the vanishing of a polynomial, or a family of polynomials, is called an algebraic set,oraZariski closed set. Algebraic geometry describes the behavior of these sets. ..."
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Let k be an algebraically closed field. The set of ordered ntuples from k is viewed as an ndimensional space; a subset described by the vanishing of a polynomial, or a family of polynomials, is called an algebraic set,oraZariski closed set. Algebraic geometry describes the behavior of these sets. The goal of this paper is a converse.
Unimodular minimal structures
 J. London Math. Soc
, 1992
"... A strongly minimal structure D is called unimodular if any two finitetoone maps with the same domain and range have the same degree; that is if/4: (/» • V is everywhere fc4tol, then kx = kc,. THEOREM. Unimodular strongly minimal structures are locally modular. This extends Zil'ber's t ..."
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Cited by 10 (2 self)
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A strongly minimal structure D is called unimodular if any two finitetoone maps with the same domain and range have the same degree; that is if/4: (/» • V is everywhere fc4tol, then kx = kc,. THEOREM. Unimodular strongly minimal structures are locally modular. This extends Zil'ber's theorem on locally finite strongly minimal sets, Urbanik's theorem on free algebras with the Steinitz property, and applies also to minimal types in N0categorical stable theories. Strongly minimal sets A strongly minimal set is a structure D such that every definable subset of D is finite or cofinite, uniformly in the parameters. For the importance of these in model theory, see [1] and [4]; relations to combinatorial geometry are discussed in [5] and [3]. We will use the existence of a theory of rank and multiplicity (Morley rank and
COMBINATORIAL GEOMETRIES OF THE FIELD EXTENSIONS
"... Abstract. We classify projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero. We investigate the firstorder theories of such geometries and pregeometries. Then we classify the algebraic combinatorial geometries of arbitrary field extensions of the transce ..."
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Abstract. We classify projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero. We investigate the firstorder theories of such geometries and pregeometries. Then we classify the algebraic combinatorial geometries of arbitrary field extensions of the transcendence degree ≥ 5 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields.
COMBINATORIAL GEOMETRIES OF FIELD EXTENSIONS JAKUB GISMATULLIN
"... In the fields of real and complex numbers, model theoretic algebraic closure coincides with relative field theoretic algebraic closure and form a pregeometry. The starting point for this work was the question, whether the above pregeometries i.e. (C, aclQ) and (R, aclQ ∩ R) (where aclQ: P(C) → P(C ..."
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In the fields of real and complex numbers, model theoretic algebraic closure coincides with relative field theoretic algebraic closure and form a pregeometry. The starting point for this work was the question, whether the above pregeometries i.e. (C, aclQ) and (R, aclQ ∩ R) (where aclQ: P(C) → P(C) is algebraic closure over Q) are isomorphic. We will give a negative answer. However in this paper we consider a more general setup. Our results are based on similar results of Evans and Hrushovski (from [1, 2]) in the case of algebraically closed fields. We work within a large algebraically closed field C. By F ̂ and F ̂ r we denote algebraic and purely inseparable closure of a field F in C. Let K ⊂ L be an arbitrary field extension and the transcendence degree of L over K is at least 3. For X ⊆ L, let aclLK(X) be K̂(X) ∩ L. We denote by G(L/K) the pregeometry (L, aclLK). The geometry G(L/K) is obtained from L \ K ̂ by factoring out the equivalence relation: x ∼ y ⇐ ⇒ K̂(x) = K̂(y). We can also transfer the closure operation aclK from G(L/K) to G(L/K): aclK(Y/∼) = aclK(Y \ K̂)/∼.
COMBINATORIAL GEOMETRIES OF FIELD EXTENSIONS JAKUB GISMATULLIN
, 903
"... Abstract. We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields. T ..."
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Abstract. We classify the algebraic combinatorial geometries of arbitrary field extensions of transcendence degree greater than 4 and describe their groups of automorphisms. Our results and proofs extend similar results and proofs by Evans and Hrushovski in the case of algebraically closed fields. The classification of projective planes in algebraic combinatorial geometries in arbitrary fields of characteristic zero will also be given.