On Diophantine definability and decidability in some infinite totally real extensions of

Author:
Alexandra Shlapentokh

Journal:
Trans. Amer. Math. Soc. **356** (2004), 3189-3207

MSC (2000):
Primary 11U05, 11U09; Secondary 03C07

DOI:
https://doi.org/10.1090/S0002-9947-03-03343-9

Published electronically:
November 4, 2003

MathSciNet review:
2052946

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a number field, and a set of its non-Archimedean primes. Then let . Let be a finite set of prime numbers. Let be the field generated by all the -th roots of unity as and . Let be the largest totally real subfield of . Then for any , there exist a number field , and a set of non-Archimedean primes of such that has density greater than , and has a Diophantine definition over the integral closure of in .

**1.**Jean-Louis Colliot-Thélène, Alexei Skorobogatov, and Peter Swinnerton-Dyer.

Double fibres and double covers: Paucity of rational points.*Acta Arithmetica*, 79:113-135, 1997. MR**98a:11081****2.**Gunther Cornelissen and Karim Zahidi.

Topology of diophantine sets: Remarks on Mazur's conjectures.

In Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, editors,*Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry*, volume 270 of*Contemporary Mathematics*, pages 253-260. American Mathematical Society, 2000. MR**2001m:11217****3.**Martin Davis.

Hilbert's tenth problem is unsolvable.*American Mathematical Monthly*, 80:233-269, 1973. MR**47:6465****4.**Martin Davis, Yurii Matijasevich, and Julia Robinson. Hilbert's tenth problem: Diophantine approximation.

Positive aspects of a negative solution.

In*Proc. Sympos. Pure Math.*, volume 28, pages 323- 378. Amer. Math. Soc., 1976. MR**55:5522****5.**Jan Denef.

Hilbert's tenth problem for quadratic rings.*Proc. Amer. Math. Soc.*, 48:214-220, 1975. MR**50:12961****6.**Jan Denef.

Diophantine sets of algebraic integers, II.*Transactions of American Mathematical Society*, 257(1):227-236, 1980. MR**81b:12031****7.**Jan Denef and Leonard Lipshitz.

Diophantine sets over some rings of algebraic integers.*Journal of London Mathematical Society*, 18(2):385-391, 1978. MR**80a:12030****8.**Barry Green, Florian Pop, and Peter Roquette.

On Rumely's local-global principle.*Jahresber. Deutsch. Math.-Verein.*, 97(2):43-74, 1996. MR**96g:11065****9.**Moshe Jarden and Aharon Razon.

Rumely's local-global principle for algebraic P C fields over rings.*Transactions of American Mathematical Society*, 350(1):55-85, 1998. MR**98d:11142****10.**Barry Mazur.

The topology of rational points.

*Experimental Mathematics*, 1(1):35-45, 1992. MR**93j:14020****11.**Barry Mazur.

Questions of decidability and undecidability in number theory.*Journal of Symbolic Logic*, 59(2):353-371, 1994. MR**96c:03091****12.**Barry Mazur.

Speculation about the topology of rational points: An update.*Asterisque*, 228:165-181, 1995. MR**96c:11068****13.**Barry Mazur.

Open problems regarding rational points on curves and varieties.

In A. J. Scholl and R. L. Taylor, editors,*Galois Representations in Arithmetic Algebraic Geometry*. Cambridge University Press, 1998, pp. 239-265. MR**2001g:14031****14.**O. T. O'Meara.*Introduction to Quadratic Forms*.

Springer Verlag, Berlin, 1973. MR**2000m:11032**(reprint)**15.**Thanases Pheidas.

Hilbert's tenth problem for a class of rings of algebraic integers.*Proceedings of American Mathematical Society*, 104(2):611-620, 1988. MR**90b:12002****16.**Thanases Pheidas.

An effort to prove that the existential theory of is undecidable.

In Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, editors,*Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry*, volume 270 of*Contemporary Mathematics*, pages 237-252. American Mathematical Society, 2000. MR**2001m:03085****17.**Bjorn Poonen.

Hilbert's Tenth Problem and Mazur's conjecture for large subrings of . To appear.**18.**Bjorn Poonen.

Using elliptic curves of rank one towards the undecidability of Hilbert's Tenth Problem over rings of algebraic integers.

In C. Fieker and D. Kohel, editors,*Algorithmic Number Theory*, volume 2369 of*Lecture Notes in Computer Science*, pages 33-42. Springer Verlag, 2002.**19.**Julia Robinson.

Definability and decision problems in arithmetic.*Journal of Symbolic Logic*, 14:98-114, 1949. MR**11:151f****20.**Julia Robinson.

The undecidability of algebraic fields and rings.*Proceedings of the American Mathematical Society*, 10:950-957, 1959. MR**22:3691****21.**Robert S. Rumely.

Arithmetic over the ring of all algebraic integers.*J. Reine Angew. Math.*, 368:127-133, 1986. MR**87i:11041****22.**Harold Shapiro and Alexandra Shlapentokh.

Diophantine relations between algebraic number fields.*Communications on Pure and Applied Mathematics*, XLII:1113-1122, 1989. MR**92b:11018****23.**Alexandra Shlapentokh.

Extension of Hilbert's tenth problem to some algebraic number fields.*Communications on Pure and Applied Mathematics*, XLII:939-962, 1989. MR**91g:11155****24.**Alexandra Shlapentokh.

Diophantine classes of holomorphy rings of global fields.*Journal of Algebra*, 169(1):139-175, 1994. MR**95h:12007****25.**Alexandra Shlapentokh.

Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator.*Inventiones Mathematicae*, 129:489-507, 1997. MR**98h:11163****26.**Alexandra Shlapentokh.

Defining integrality at prime sets of high density in number fields.*Duke Mathematical Journal*, 101(1):117-134, 2000. MR**2001a:11200****27.**Alexandra Shlapentokh.

Hilbert's tenth problem over number fields, a survey.

In Jan Denef, Leonard Lipshitz, Thanases Pheidas, and Jan Van Geel, editors,*Hilbert's Tenth Problem: Relations with Arithmetic and Algebraic Geometry*, volume 270 of*Contemporary Mathematics*, pages 107-137. American Mathematical Society, 2000. MR**2001m:03023****28.**Alexandra Shlapentokh.

On diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2.*Journal of Number Theory*, 95:227-252, 2002. MR**2003h:03068****29.**Alexandra Shlapentokh.

A ring version of Mazur's conjecture on topology of rational points.*International Mathematics Research Notices*, 2003:7:411-423, 2003.

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Additional Information

**Alexandra Shlapentokh**

Affiliation:
Department of Mathematics, East Carolina University, Greenville, North Carolina 27858

Email:
shlapentokha@mail.ecu.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03343-9

Keywords:
Hilbert's tenth problem,
Diophantine definability

Received by editor(s):
June 5, 2000

Received by editor(s) in revised form:
March 10, 2003

Published electronically:
November 4, 2003

Additional Notes:
The research for this paper has been partially supported by NSA grant MDA904-98-1-0510 and NSF grant DMS-9988620

Article copyright:
© Copyright 2003
American Mathematical Society