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On Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2
, 2001
"... Let M be a number field. Let W be a set of nonarchimedean primes of M . Let OM,W = {x # M  ord p x # 0 #p ## W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper she improves her previous density estimat ..."
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Let M be a number field. Let W be a set of nonarchimedean primes of M . Let OM,W = {x # M  ord p x # 0 #p ## W}. The author continues her investigation of Diophantine definability and decidability in rings OM,W where W is infinite. In this paper she improves her previous density estimates and extends the results to the totally complex extensions of degree 2 of the totally real fields. In particular, the following results are proved. Let M be a totally real field or a totally complex extension of degree 2 of a totally real field. Then for any # > 0 there exists a set WM of primes of M whose density is bigger than 1  [M : Q] 1  # and such that Z has a Diophantine definition over OM,WM . (Thus, Hilbert's Tenth Problem is undecidable in OM,WM .) Let M be as above and let # > 0 be given. Let SQ be the set of all rational primes splitting in M . (If the extension is Galois but not cyclic, SQ contains all the rational primes.) Then there exists a set of Mprimes WM such that the set of rational primes WQ below WM di#ers from SQ by a set contained in a set of density less than # and such that Z has a Diophantine definition over OM,WM . (Again this will imply that Hilbert's Tenth Problem is undecidable in OM,WM .) 1
Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers
"... Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of . ..."
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Cited by 12 (2 self)
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Let F be their rings of integers. If there exists an elliptic curve E over F such that rk E(F ) = rk E(K) = 1, then there exists a diophantine definition of .
Hilbert’s tenth problem for algebraic function fields over infinite fields of constants of positive characteristic
 Pacific Journal of Mathematics
, 2000
"... Let K be an algebraic function field of characteristic p>2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfield K1, possibly equal to C, and elements u, x such that u is transcendental over K1, x is algebraic over ..."
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Cited by 10 (4 self)
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Let K be an algebraic function field of characteristic p>2. Let C be the algebraic closure of a finite field in K. Assume that C has an extension of degree p. Assume also that K contains a subfield K1, possibly equal to C, and elements u, x such that u is transcendental over K1, x is algebraic over C(u) and K = K1(u, x). Then the Diophantine problem of K is undecidable. Let G be an algebraic function field in one variable whose constant field is algebraic over a finite field and is not algebraically closed. Then for any prime p of G, the set of elements of G integral at p is Diophantine over G. 1. Introduction. The interest in the questions of Diophantine definability and decidability goes back to a question which was posed by Hilbert: Given an arbitrary polynomial equation in several variables over Z, is there a uniform algorithm
DEFINING INTEGRALITY AT PRIME SETS OF HIGH DENSITY IN NUMBER FIELDS
 VOL. 101, NO. 1 DUKE MATHEMATICAL JOURNAL
, 2000
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A ring version of Mazur’s conjecture on topology of rational points
 Internat. Math. Res. Notices
"... The purpose of this paper is to explore a conjecture due to Barry Mazur and formulated ..."
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Cited by 10 (7 self)
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The purpose of this paper is to explore a conjecture due to Barry Mazur and formulated
On diophantine definability and decidability in some infinite totally real extensions
 of Q, Trans. Amer. Math. Soc
"... Abstract. Let M be a number field, and WM a set of its nonArchimedean primes. Then let OM,W = {x ∈ M  ordt x ≥ 0, ∀t � ∈ WM}. Let{p1,...,pr} M be a finite set of prime numbers. Let Finf be the field generated by all the p j ith roots of unity as j → ∞ and i = 1,...,r. Let Kinf be the largest tot ..."
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Abstract. Let M be a number field, and WM a set of its nonArchimedean primes. Then let OM,W = {x ∈ M  ordt x ≥ 0, ∀t � ∈ WM}. Let{p1,...,pr} M be a finite set of prime numbers. Let Finf be the field generated by all the p j ith roots of unity as j → ∞ and i = 1,...,r. Let Kinf be the largest totally real subfield of Finf. Then for any ε>0, there exist a number field M ⊂ Kinf,andasetWMof nonArchimedean primes of M such that WM has density greater than 1 − ε, andZhas a Diophantine definition over the integral closure of OM,W in Kinf. M 1.
ELLIPTIC CURVES RETAINING THEIR RANK IN FINITE EXTENSIONS AND HILBERT’S TENTH PROBLEM FOR RINGS OF ALGEBRAIC NUMBERS
, 2008
"... Abstract. Using Poonen’s version of the “weak vertical method ” we produce new examples of “large ” and “small ” rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture ..."
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Abstract. Using Poonen’s version of the “weak vertical method ” we produce new examples of “large ” and “small ” rings of algebraic numbers (including rings of integers) where Z and/or the ring of integers of a subfield are existentially definable and/or where the ring version of Mazur’s conjecture on the topology of rational points does not hold. 1.
Diophantine Undecidability of Function Fields of Characteristic Greater Than 2, Finitely Generated over Fields Algebraic over a Finite Field
, 2001
"... Let F be a function field of characteristic p>2, finitely generated over a field C algebraic over a finite field Cp and such that it has an extension of degree p. Then Hilbert's Tenth Problem is not decidable over F . 1 ..."
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Let F be a function field of characteristic p>2, finitely generated over a field C algebraic over a finite field Cp and such that it has an extension of degree p. Then Hilbert's Tenth Problem is not decidable over F . 1
DIOPHANTINE DEFINABILITY AND DECIDABILITY IN THE EXTENSIONS OF DEGREE 2 OF TOTALLY REAL FIELDS
, 2006
"... Abstract. We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown ..."
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Abstract. We investigate Diophantine definability and decidability over some subrings of algebraic numbers contained in quadratic extensions of totally real algebraic extensions of Q. Among other results we prove the following. The big subring definability and undecidability results previously shown by the author to hold over totally complex extensions of degree 2 of totally real number fields, are shown to hold for all extensions of degree 2 of totally real number fields. The definability and undecidability results for integral closures of “small ” and “big ” subrings of number fields in the infinite algebraic extensions of Q, previously shown by the author to hold for totally real fields, are extended to a large class of extensions of degree 2 of totally real fields. This class includes infinite cyclotomics and abelian extensions with finitely many ramified rational primes. 1.
Hilbert's Tenth Problem over Rings of NumberTheoretic Interest
, 2003
"... This article is a survey about analogues of Hilbert's Tenth Problem over various rings, especially rings of interest to number theorists and algebraic geometers. For more details about most of the topics considered here, the conference proceedings [DLPVG00] is recommended. 2. The original problem H ..."
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This article is a survey about analogues of Hilbert's Tenth Problem over various rings, especially rings of interest to number theorists and algebraic geometers. For more details about most of the topics considered here, the conference proceedings [DLPVG00] is recommended. 2. The original problem Hilbert's Tenth Problem (from his list of 23 problems published in 1900) asked for an algorithm to decide whether a diophantine equation has a solution. More precisely, the input and output of such an algorithm were to be as follows: input: a polynomial f(x 1 , . . . , x n ) having coe#cients in Z Date: February 28, 2003