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  1.  7
    A Computable Functor From Graphs to Fields.Russell Miller, Bjorn Poonen, Hans Schoutens & Alexandra Shlapentokh - 2018 - Journal of Symbolic Logic 83 (1):326-348.
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  2. Weak Presentations of Computable Fields.Carl G. Jockusch & Alexandra Shlapentokh - 1995 - Journal of Symbolic Logic 60 (1):199 - 208.
    It is shown that for any computable field K and any r.e. degree a there is an r.e. set A of degree a and a field F ≅ K with underlying set A such that the field operations of F (including subtraction and division) are extendible to (total) recursive functions. Further, it is shown that if a and b are r.e. degrees with b ≤ a, there is a 1-1 recursive function $f: \mathbb{Q} \rightarrow \omega$ such that f(Q) ∈ a, (...)
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  3.  29
    Defining Integers.Alexandra Shlapentokh - 2011 - Bulletin of Symbolic Logic 17 (2):230-251.
    This paper surveys the recent developments in the area that grew out of attempts to solve an analog of Hilbert's Tenth Problem for the field of rational numbers and the rings of integers of number fields. It is based on a plenary talk the author gave at the annual North American meeting of ASL at the University of Notre Dame in May of 2009.
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  4.  6
    Rings of Algebraic Numbers in Infinite Extensions of $${\Mathbb {Q}}$$ and Elliptic Curves Retaining Their Rank.Alexandra Shlapentokh - 2009 - Archive for Mathematical Logic 48 (1):77-114.
    We show that elliptic curves whose Mordell–Weil groups are finitely generated over some infinite extensions of ${\mathbb {Q}}$ , can be used to show the Diophantine undecidability of the rings of integers and bigger rings contained in some infinite extensions of rational numbers.
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  5.  10
    Diophantine Equivalence and Countable Rings.Alexandra Shlapentokh - 1994 - Journal of Symbolic Logic 59 (3):1068-1095.
    We show that Diophantine equivalence of two suitably presented countable rings implies that the existential polynomial languages of the two rings have the same "expressive power" and that their Diophantine sets are in some sense the same. We also show that a Diophantine class of countable rings is contained completely within a relative enumeration class and demonstrate that one consequence of this fact is the existence of infinitely many Diophantine classes containing holomophy rings of Q.
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  6.  3
    Rational Separability Over a Global Field.Alexandra Shlapentokh - 1996 - Annals of Pure and Applied Logic 79 (1):93-108.
    Let F be a finitely generated field and let j : F → N be a weak presentation of F, i.e. an isomorphism from F onto a field whose universe is a subset of N and such that all the field operations are extendible to total recursive functions. Then if R1 and R2 are recursive subrings of F, for all weak presentations j of F, j is Turing reducible to j if and only if there exists a finite collection of (...)
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  7.  6
    Diophantine Undecidability in Some Rings of Algebraic Numbers of Totally Real Infinite Extensions of Q.Alexandra Shlapentokh - 1994 - Annals of Pure and Applied Logic 68 (3):299-325.
    This paper provides the first examples of rings of algebraic numbers containing the rings of algebraic integers of the infinite algebraic extensions of where Hilbert's Tenth Problem is undecidable.
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  8.  3
    A Diophantine Definition of Rational Integers Over Some Rings of Algebraic Numbers.Alexandra Shlapentokh - 1992 - Notre Dame Journal of Formal Logic 33 (3):299-321.
  9.  22
    Diophantine Relations Between Rings of s-Integers of Fields of Algebraic Functions in One Variable Over Constant Fields of Positive Characteristic.Alexandra Shlapentokh - 1993 - Journal of Symbolic Logic 58 (1):158-192.
    One of the main theorems of the paper states the following. Let R-K-M be finite extensions of a rational one variable function field R over a finite field of constants. Let S be a finite set of valuations of K. Then the ring of elements of K having no poles outside S has a Diophantine definition over its integral closure in M.
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  10.  5
    Weak Presentations of Non-Finitely Generated Fields.Alexandra Shlapentokh - 1998 - Annals of Pure and Applied Logic 94 (1-3):223-252.
    Let K be a countable field. Then a weak presentation of K is an isomorphism of K onto a field whose elements are natural numbers, such that all the field operations are extendible to total recursive functions. Given a pair of two non-finitely generated countable fields contained in some overfield, we investigate under what circumstances the overfield has a weak presentation under which the given fields have images of arbitrary Turing degrees or, in other words, we investigate Turing separability of (...)
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  11.  8
    Decidable Algebraic Fields.Moshe Jarden & Alexandra Shlapentokh - 2017 - Journal of Symbolic Logic 82 (2):474-488.
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  12. Notre Dame, Indiana May 20–May 23, 2009.Patricia Blanchette, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber - 2009 - Bulletin of Symbolic Logic 15 (4).
     
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  13.  7
    Diophantine Definability Over Non-Finitely Generated Non-Degenerate Modules of Algebraic Extensions of ℚ.Alexandra Shlapentokh - 2001 - Archive for Mathematical Logic 40 (4):297-328.
    We investigate the issues of Diophantine definability over the non-finitely generated version of non-degenerate modules contained in the infinite algebraic extensions of the rational numbers. In particular, we show the following. Let k be a number field and let K inf be a normal algebraic, possibly infinite, extension of k such that k has a normal extension L linearly disjoint from K inf over k. Assume L is totally real and K inf is totally complex. Let M inf be a (...)
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  14.  7
    Generalized Weak Presentations.Alexandra Shlapentokh - 2002 - Journal of Symbolic Logic 67 (2):787-819.
    Let K be a computable field. Let F be a collection of recursive functions over K, possibly including field operations. We investigate the following question. Given an r.e. degree a, is there an injective map j: K $\longrightarrow \mathbb{N}$ such that j(K) is of degree a and all the functions in F are translated by restrictions of total recursive functions.
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  15.  5
    First-Order Definitions of Rational Functions and S -Integers Over Holomorphy Rings of Algebraic Functions of Characteristic 0.Alexandra Shlapentokh - 2005 - Annals of Pure and Applied Logic 136 (3):267-283.
    We consider the problem of constructing first-order definitions in the language of rings of holomorphy rings of one-variable function fields of characteristic 0 in their integral closures in finite extensions of their fraction fields and in bigger holomorphy subrings of their fraction fields. This line of questions is motivated by similar existential definability results over global fields and related questions of Diophantine decidability.
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