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  1. Alexandra Shlapentokh (2011). Defining Integers. 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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  2. Patricia Blanchette, Kit Fine, Heike Mildenberger, André Nies, Anand Pillay, Alexander Razborov, Alexandra Shlapentokh, John R. Steel & Boris Zilber (2009). Notre Dame, Indiana May 20–May 23, 2009. Bulletin of Symbolic Logic 15 (4).
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  3. Alexandra Shlapentokh (2009). Rings of Algebraic Numbers in Infinite Extensions of {Mathbb {Q}} and Elliptic Curves Retaining Their Rank. 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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  4. Alexandra Shlapentokh (2005). First-Order Definitions of Rational Functions and -Integers Over Holomorphy Rings of Algebraic Functions of Characteristic 0. Annals of Pure and Applied Logic 136 (3):267-283.
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  5. Alexandra Shlapentokh (2003). Existential Definability with Bounds on Archimedean Valuations. Journal of Symbolic Logic 68 (3):860-878.
    We show that a solution to Hilbert's Tenth Problem in the rings of algebraic integers and bigger subrings of number fields where it is currently not known, is equivalent to a problem of bounding archimedean valuations over non-real number fields.
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  6. Alexandra Shlapentokh (2002). Generalized Weak Presentations. 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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  7. Alexandra Shlapentokh (2002). On Diophantine Definability and Decidability in Some Rings of Algebraic Functions of Characteristic. Journal of Symbolic Logic 67 (2):759-786.
    Let K be a function field of one variable over a constant field C of finite transcendence degree over C. Let M/K be a finite extension and let W be a set of primes of K such that all but finitely many primes of W do not split in the extension M/K. Then there exists a set W' of K-primes such that Hilbert's Tenth Problem is not decidable over $O_{K,W'} = \{x \in K\mid ord_\mathfrak{p} x \geq 0, \forall\mathfrak{p} \notin W'\}$ (...)
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  8. Alexandra Shlapentokh (2001). Diophantine Definability Over Non-Finitely Generated Non-Degenerate Modules of Algebraic Extensions of ℚ. 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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  9. Alexandra Shlapentokh (1998). Weak Presentations of Non-Finitely Generated Fields. Annals of Pure and Applied Logic 94 (1-3):223-252.
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  10. Alexandra Shlapentokh (1996). Rational Separability Over a Global Field. Annals of Pure and Applied Logic 79 (1):93-108.
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  11. Carl G. Jockusch Jr & Alexandra Shlapentokh (1995). Weak Presentations of Computable Fields. 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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  12. Alexandra Shlapentokh (1994). Diophantine Equivalence and Countable Rings. 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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  13. Alexandra Shlapentokh (1994). Diophantine Undecidability in Some Rings of Algebraic Numbers of Totally Real Infinite Extensions Of. Annals of Pure and Applied Logic 68 (3):299-325.
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  14. Alexandra Shlapentokh (1993). Diophantine Relations Between Rings of s-Integers of Fields of Algebraic Functions in One Variable Over Constant Fields of Positive Characteristic. 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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  15. Alexandra Shlapentokh (1992). A Diophantine Definition of Rational Integers Over Some Rings of Algebraic Numbers. Notre Dame Journal of Formal Logic 33 (3):299-321.
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