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A note on groups definable in difference fields
 Proceedings of the AMS, 130 (2001), 205
, 2000
"... In this paper we record some observations around groups definable in difference fields. We were motivated by a question of Zoe Chatzidakis as to whether any group definable in a model of ACF A is virtually definably embeddable in an algebraic group. We give a positive answer using routine ..."
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In this paper we record some observations around groups definable in difference fields. We were motivated by a question of Zoe Chatzidakis as to whether any group definable in a model of ACF A is virtually definably embeddable in an algebraic group. We give a positive answer using routine
Some modeltheoretic and geometric properties of fields with jet operators
, 2001
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ITERATIVE DIFFERENTIAL GALOIS THEORY IN POSITIVE CHARACTERISTIC: A MODEL THEORETIC APPROACH
"... Abstract. This paper introduces a natural extension of Kolchin’s differential Galois theory to positive characteristic iterative differential fields, generalizing to the nonlinear case the iterative PicardVessiot theory recently developed by Matzat and van der Put. We use the methods and framework ..."
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Abstract. This paper introduces a natural extension of Kolchin’s differential Galois theory to positive characteristic iterative differential fields, generalizing to the nonlinear case the iterative PicardVessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal extension of iterative differential fields, and then prove that these extensions have good Galois theory and that a Gprimitive element theorem holds. In addition, making use of the basic theory of arc spaces of algebraic groups, we define iterative logarithmic equations, finally proving that our strongly normal extensions are Galois extensions for these equations. 1.
A survey of the uncountable spectra of countable theories
 in Algebraic Model Theory
, 1996
"... Let T be a complete, firstorder theory in a finite or countable language having infinite models. Let I(T,κ) be the number of isomorphism types of models of T of cardinality κ. We denote by µ (respectively ˆµ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. Theorem ..."
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Let T be a complete, firstorder theory in a finite or countable language having infinite models. Let I(T,κ) be the number of isomorphism types of models of T of cardinality κ. We denote by µ (respectively ˆµ) the number of cardinals (respectively infinite cardinals) less than or equal to κ. Theorem. I(T,κ), as a function of κ> ℵ0, is the minimum of 2κ and one of the following functions:
Countable imaginary simple unidimensional theories are supersimple
, 2008
"... We prove that a countable simple unidimensional theory that eliminates hyperimaginaries is supersimple. This solves a problem of Shelah in the more general context of simple theories under weak assumptions. 1 ..."
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We prove that a countable simple unidimensional theory that eliminates hyperimaginaries is supersimple. This solves a problem of Shelah in the more general context of simple theories under weak assumptions. 1