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Sergei Starchenko [14]S. Starchenko [7]
  1.  49
    Vapnik–Chervonenkis Density in Some Theories without the Independence Property, II.Matthias Aschenbrenner, Alf Dolich, Deirdre Haskell, Dugald Macpherson & Sergei Starchenko - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):311-363.
    We study the Vapnik–Chervonenkis density of definable families in certain stable first-order theories. In particular, we obtain uniform bounds on the VC density of definable families in finite $\mathrm {U}$-rank theories without the finite cover property, and we characterize those abelian groups for which there exist uniform bounds on the VC density of definable families.
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  2.  36
    Definable homomorphisms of abelian groups in o-minimal structures.Ya'acov Peterzil & Sergei Starchenko - 1999 - Annals of Pure and Applied Logic 101 (1):1-27.
    We investigate the group of definable homomorphisms between two definable abelian groups A and B, in an o-minimal structure . We prove the existence of a “large”, definable subgroup of . If contains an infinite definable set of homomorphisms then some definable subgroup of B admits a definable multiplication, making it into a field. As we show, all of this can be carried out not only in the underlying structure but also in any structure definable in.
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  3.  30
    Groups Definable in Ordered Vector Spaces over Ordered Division Rings.Pantelis E. Eleftheriou & Sergei Starchenko - 2007 - Journal of Symbolic Logic 72 (4):1108 - 1140.
    Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G, ⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably connected with respect to the t-topology, then it is definably isomorphic to a 'definable quotient group' U/L, for some convex V-definable subgroup U of 〈Mⁿ, +〉 and a lattice L of rank n. As two consequences, we derive Pillay's conjecture (...)
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  4.  33
    Forking in VC-minimal theories.Sarah Cotter & Sergei Starchenko - 2012 - Journal of Symbolic Logic 77 (4):1257-1271.
    We consider VC-minimal theories admitting unpackable generating families, and show that in such theories, forking of formulae over a model M is equivalent to containment in global types definable over M, generalizing a result of Dolich on o-minimal theories in [4].
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  5.  30
    On forking and definability of types in some dp-minimal theories.Pierre Simon & Sergei Starchenko - 2014 - Journal of Symbolic Logic 79 (4):1020-1024.
    We prove in particular that, in a large class of dp-minimal theories including the p-adics, definable types are dense amongst nonforking types.
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  6.  47
    Real closed fields and models of Peano arithmetic.P. D'Aquino, J. F. Knight & S. Starchenko - 2010 - Journal of Symbolic Logic 75 (1):1-11.
    Shepherdson [14] showed that for a discrete ordered ring I, I is a model of IOpen iff I is an integer part of a real closed ordered field. In this paper, we consider integer parts satisfying PA. We show that if a real closed ordered field R has an integer part I that is a nonstandard model of PA (or even IΣ₄), then R must be recursively saturated. In particular, the real closure of I, RC (I), is recursively saturated. We (...)
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  7.  50
    Geometry, calculus and Zil'ber's conjecture.Ya'acov Peterzil & Sergei Starchenko - 1996 - Bulletin of Symbolic Logic 2 (1):72-83.
    §1. Introduction. By and large, definitions of a differentiable structure on a set involve two ingredients, topology and algebra. However, in some cases, partial information on one or both of these is sufficient. A very simple example is that of the field ℝ where algebra alone determines the ordering and hence the topology of the field:In the case of the field ℂ, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the ield but (...)
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  8.  34
    Expansions of algebraically closed fields II: Functions of several variables.Ya'acov Peterzil & Sergei Starchenko - 2003 - Journal of Mathematical Logic 3 (01):1-35.
    Let ℛ be an o-minimal expansion of a real closed field R. We continue here the investigation we began in [11] of differentiability with respect to the algebraically closed field [Formula: see text]. We develop the basic theory of such K-differentiability for definable functions of several variables, proving theorems on removable singularities as well as analogues of the Weierstrass preparation and division theorems for definable functions. We consider also definably meromorphic functions and prove that every definable function which is meromorphic (...)
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  9.  26
    Introduction.Zoé Chatzidakis, David Marker, Amador Martin-Pizarro, Rahim Moosa & Sergei Starchenko - 2013 - Notre Dame Journal of Formal Logic 54 (3-4):277-277.
    Zoé Chatzidakis , David Marker , Amador Martin-Pizarro , Rahim Moosa , Sergei Starchenko Source: Notre Dame J. Formal Logic, Volume 54, Number 3-4, 277--277.
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  10.  26
    Corrigendum to: “Real closed fields and models of arithmetic”.P. D'Aquino, J. F. Knight & S. Starchenko - 2012 - Journal of Symbolic Logic 77 (2):726-726.
  11.  30
    San Antonio Convention Center San Antonio, Texas January 14–15, 2006.Douglas Cenzer, C. Ward Henson, Michael C. Laskowski, Alain Louveau, Russell Miller, Itay Neeman, Sergei Starchenko & Valentina Harizanov - 2006 - Bulletin of Symbolic Logic 12 (4).
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  12.  12
    Model-theoretic Elekes–Szabó in the strongly minimal case.Artem Chernikov & Sergei Starchenko - 2020 - Journal of Mathematical Logic 21 (2):2150004.
    We prove a generalization of the Elekes–Szabó theorem [G. Elekes and E. Szabó, How to find groups?, Combinatorica 32 537–571 ] for relations defina...
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  13. Real closed fields and models of arithmetic (vol 75, pg 1, 2010).P. D'Aquino, J. F. Knight & S. Starchenko - 2012 - Journal of Symbolic Logic 77 (2).
  14.  37
    Madison, WI, USA March 31–April 3, 2012.Alan Dow, Isaac Goldbring, Warren Goldfarb, Joseph Miller, Toniann Pitassi, Antonio Montalbán, Grigor Sargsyan, Sergei Starchenko & Moshe Vardi - 2013 - Bulletin of Symbolic Logic 19 (2).
  15.  27
    Addendum to "a structure theorem for strongly Abelian varieties".Bradd Hart & Sergei Starchenko - 1993 - Journal of Symbolic Logic 58 (4):1419-1425.
  16.  42
    1-based theories — the main gap for a -models.B. Hart, A. Pillay & S. Starchenko - 1995 - Archive for Mathematical Logic 34 (5):285-300.
    We prove the Main Gap for the class of a -models (sufficiently saturated models) of an arbitrary stable 1-based theory T . We (i) prove a strong structure theorem for a -models, assuming NDOP, and (ii) roughly compute the number of a -models of T in any given cardinality. The analysis uses heavily group existence theorems in 1-based theories.
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  17.  10
    1-based theories - the main gap for $a$ -models.B. Hart, A. Pillay & S. Starchenko - 1995 - Archive for Mathematical Logic 34 (5):285-300.
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  18.  12
    Superstable quasi-varieties.B. Hart & S. Starchenko - 1994 - Annals of Pure and Applied Logic 69 (1):53-71.
    We present a structure theorem for superstable quasi-varieties without DOP. We show that every algebra in such a quasi-variety weakly decomposes as the product of an affine algebra and a combinational algebra, that is, it is bi-interpretable with a two sorted structure where one sort is an affine algebra, the other sort is a combinatorial algebra and the only non-trivial polynomials between the two sorts are certain actions of the affine sort on the combinatorial sort.
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  19.  22
    Triviality, NDOP and stable varieties.B. Hart, A. Pillay & S. Starchenko - 1993 - Annals of Pure and Applied Logic 62 (2):119-146.
    We study perfectly trivial theories, 1-based theories, stable varieties, and their mutual interaction. We give a structure theorem for the models of a complete perfectly trivial stable theory without DOP: any model is the algebraic closure of a nonforking regular tree of elements. We also give a structure theorem for stable varieties, all of whose completions have NDOP. Such a variety is a varietal product of an affine variety and a combinatorial variety of an especially simple form.
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