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31
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 18 (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 .
Open Diophantine Problems
 MOSCOW MATHEMATICAL JOURNAL
, 2004
"... Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendent ..."
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Cited by 15 (3 self)
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Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.
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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Cited by 13 (11 self)
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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
DEFINING INTEGRALITY AT PRIME SETS OF HIGH DENSITY IN NUMBER FIELDS
 VOL. 101, NO. 1 DUKE MATHEMATICAL JOURNAL
, 2000
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DIOPHANTINE DEFINABILITY OF INFINITE DISCRETE NONARCHIMEDEAN SETS AND DIOPHANTINE MODELS OVER Large Subrings Of Number Fields
, 2004
"... We prove that infinite padically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree2 extension of a totally real number field, then there exists a prime p of K and a set of Kprimes S of densit ..."
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We prove that infinite padically discrete sets have Diophantine definitions in large subrings of some number fields. First, if K is a totally real number field or a totally complex degree2 extension of a totally real number field, then there exists a prime p of K and a set of Kprimes S of density arbitrarily close to 1 such that there is an infinite padically discrete set that is Diophantine over the ring OK,S of Sintegers in K. Second, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of Kprimes S of density 1 and an infinite Diophantine subset of OK,S that is vadically discrete for every place v of K. Third, if K is a number field over which there exists an elliptic curve of rank 1, then there exists a set of Kprimes S of density 1 such that there exists a Diophantine model of Z over OK,S. This line of research is motivated by a question of Mazur concerning the distribution of rational points on varieties in a nonarchimedean topology and questions concerning extensions of Hilbert’s Tenth Problem to subrings of number fields.
Extensions of Büchi’s problem : Questions of decidability for addition and kth powers
"... The authors thank the referee for his comments. The second author aknowledges the hospitality of the University of CreteHeraklion, where the main part of this work was done. Abstract. We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algo ..."
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The authors thank the referee for his comments. The second author aknowledges the hospitality of the University of CreteHeraklion, where the main part of this work was done. Abstract. We generalize a question of Büchi: Let R be an integral domain, C a subring and k ≥ 2 an integer. Is there an algorithm to decide the solvability in R of any given system of polynomial equations, each of which is linear in the k−th powers of the unknowns, with coefficients in C? We state a numbertheoretical problem, depending on k, a positive answer to which would imply a negative answer to the question for R = C = Z. We reduce a negative answer for k = 2 and for R = F (t), a field of rational functions of zero characteristic, to the undecidability of the ring theory of F (t). We address the similar question, where we allow, along with the equations, also conditions of the form ‘x is a constant ’ and ‘x takes the value 0 at t = 0’, for k = 3 and for function fields R = F (t) of zero characteristic, with C = Z[t]. We prove that a negative answer to this question would follow from a negative answer for a ring between Z and the extension of Z by a primitive cube root of 1.
COUNTING SPECIAL POINTS: LOGIC, DIOPHANTINE GEOMETRY AND TRANSCENDENCE THEORY
"... ABSTRACT. We expose a theorem of Pila and Wilkie on counting rational points in sets definable in ominimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier. 1. ..."
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ABSTRACT. We expose a theorem of Pila and Wilkie on counting rational points in sets definable in ominimal structures and some applications of this theorem to problems in diophantine geometry due to Masser, Peterzil, Pila, Starchenko, and Zannier. 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.
THE ANALOGUE OF BÜCHI’S PROBLEM FOR RATIONAL FUNCTIONS
, 2006
"... Büchi’s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form x 2 n + x2n−2 =2x2n−1 +2, n =2,...,M − 1, has no integer points other than those that satisfy ±xn = ±x0 + n (the ± signs are independent). If answered positively, it would imply ..."
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Cited by 6 (2 self)
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Büchi’s problem asked whether there exists an integer M such that the surface defined by a system of equations of the form x 2 n + x2n−2 =2x2n−1 +2, n =2,...,M − 1, has no integer points other than those that satisfy ±xn = ±x0 + n (the ± signs are independent). If answered positively, it would imply that there is no algorithm which decides, given an arbitrary system Q =(q1,...,qr) of integral quadratic forms and an arbitrary rtuple B =(b1,...,br) of integers, whether Q represents B (see T. Pheidas and X. Vidaux, Fund. Math. 185 (2005) 171–194). Thus it would imply the following strengthening of the negative answer to Hilbert’s tenth problem: the positiveexistential theory of the rational integers in the language of addition and a predicate for the property ‘x is a square ’ would be undecidable. Despite some progress, including a conditional positive answer (depending on conjectures of Lang), Büchi’s problem remains open. In this paper we prove the following: (A) an analogue of Büchi’s problem in rings of polynomials of characteristic either 0 or p � 17 and for fields of rational functions of characteristic 0; and (B) an analogue of Büchi’s problem in fields of rational functions of characteristic p � 19, but only for sequences that satisfy a certain additional hypothesis. As a consequence we prove the following result in logic. Let F be a field of characteristic either 0 or at least 17 and let t be a variable. Let Lt be the first order language which contains symbols for 0 and 1, a symbol for addition, a symbol for the property ‘x is a square ’ and symbols for multiplication by each element of the image of Z[t] in F [t]. Let R beasubringofF (t), containing the natural image of Z[t] inF (t). Assume that one of the following is true: (i) R ⊂ F [t]; (ii) the characteristic of F is either 0 or p � 19. Then multiplication is positiveexistentially definable over the ring R, in the language Lt. Hence the positiveexistential theory of R in Lt is decidable if and only if the positiveexistential ringtheory of R in the language of rings, augmented by a constantsymbol for t, is decidable. 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.