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Defining Integers

Published online by Cambridge University Press:  15 January 2014

Alexandra Shlapentokh
Alexandra Shlapentokh*
Affiliation:
Department of Mathematics, East Carolina University, Greenville, NC 27858, USAE-mail:[email protected], URL: www.personal.ecu.edu/shlapentokha
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Abstract

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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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 17 , Issue 2 , June 2011 , pp. 230 - 251
Copyright
Copyright © Association for Symbolic Logic 2011

References

REFERENCES

[1] Cornelissen, Gunther, Pheidas, Thanases, and Zahidi, Karim, Division-ample sets and diophantine problem for rings of integers, Journal de Théorie des Nombres Bordeaux, vol. 17 (2005), pp. 727–735.Google Scholar
[2] Cornelissen, Gunther and Shlapentokh, Alexandra, Defining the integers in large rings of number fields using one universal quantifier, Zapiski Nauchnykh Seminarov POMI, vol. 358 (2008), pp. 199–223.Google Scholar
[3] Cornelissen, Gunther and Zahidi, Karim, Topology of diophantine sets: Remarks on Mazur's conjectures, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry (Denef, Jan, Lipshitz, Leonard, Pheidas, Thanases, and Van Geel, Jan, editors), Contemporary Mathematics, vol. 270, American Mathematical Society, 2000, pp. 253–260.Google Scholar
[4] Cornelissen, Gunther and Zahidi, Karim, Elliptic divisibility sequences and undecidable problems about rational points, Journal für die Reine und Angewandte Mathematik, vol. 613 (2007), pp. 1–33.Google Scholar
[5] Davis, Martin, Hilbert's tenth problem is unsolvable, American Mathematical Monthly, vol. 80 (1973), pp. 233–269.Google Scholar
[6] Davis, Martin, Matiyasevich, Yuri, and Robinson, Julia, Hilbert's tenth problem, diophantine equations: Positive aspects of a negative solution, Proceedings of the symposium on pure mathematics, vol. 28, American Mathematical Society, 1976, pp. 323–378.CrossRefGoogle Scholar
[7] Denef, Jan, Hilbert's tenth problem for quadratic rings, Proceedings of the American Mathematical Society, vol. 48 (1975), pp. 214–220.Google Scholar
[8] Denef, Jan and Lipshitz, Leonard, Diophantine sets over some rings of algebraic integers, Journal of London Mathematical Society, vol. 18 (1978), no. 2, pp. 385–391.CrossRefGoogle Scholar
[9] Denef, Jan, Lipshitz, Leonard, Pheidas, Thanases, and Van Geel, Jan (editors), Hilbert's tenth problem: Relations with arithmetic and algebraic geometry, Contemporary Mathematics, vol. 270, American Mathematical Society, Providence, RI, 2000.Google Scholar
[10] Eisenträger, Kirsten and Everest, Graham, Descent on elliptic curves and Hilbert's tenth problem, Proceedings of the American Mathematical Society, vol. 137 (2009), no. 6, pp. 1951–1959.Google Scholar
[11] Everest, Graham, van der Poorten, Alf, Shparlinski, Igor, and Ward, Thomas, Recurrence sequences. Mathematical Surveys and Monographs, vol. 104, American Mathematical Society, Providence, RI, 2003.Google Scholar
[12] Marker, David, Model theory, Graduate Texts in Mathematics, vol. 217, Springer-Verlag, New York, 2002, an introduction.Google Scholar
[13] Matiyasevich, Yuri V., Hilbert's tenth problem, Foundations of Computing Series, MIT Press, Cambridge, MA, 1993, translated from the 1993 Russian original by the author.Google Scholar
[14] Mazur, Barry, The topology of rational points, Experimental Mathematics, vol. 1 (1992), no. 1, pp. 35–45.CrossRefGoogle Scholar
[15] Mazur, Barry, Questions of decidability and undecidability in number theory, The Journal of Symbolic Logic, vol. 59 (1994), no. 2, pp. 353–371.Google Scholar
[16] Mazur, Barry, Open problems regarding rational points on curves and varieties, Galois representations in arithmetic algebraic geometry (Scholl, A. J. and Taylor, R. L., editors), Cambridge University Press, 1998.Google Scholar
[17] Mazur, Barry and Rubin, Karl, Ranks of twists of elliptic curves and Hilbert's tenth problem, Inventiones Mathematicae, vol. 181 (2010), pp. 541–575.Google Scholar
[18] Pheidas, Thanases, Hilbert's tenth problem for a class ofrings of algebraic integers, Proceedings of American Mathematical Society, vol. 104 (1988), no. 2, pp. 611–620.Google Scholar
[19] Poonen, Bjorn, Using elliptic curves of rank one towards the undecidability of Hilbert's tenth problem over rings of algebraic integers, Algorithmic number theory (Fieker, C. and Kohel, D., editors), Lecture Notes in Computer Science, vol. 2369, Springer Verlag, 2002, pp. 33–42.CrossRefGoogle Scholar
[20] Poonen, Bjorn, Hilbert's tenth problem and Mazur's conjecture for large subrings of ℚ, Journal of the American Mathematical Society, vol. 16 (2003), no. 4, pp. 981–990.CrossRefGoogle Scholar
[21] Poonen, Bjorn, Undecidability in number theory, Notices in the American Mathematical Society, vol. 55 (2008), no. 3, pp. 344–350.Google Scholar
[22] Poonen, Bjorn, Characterizing integers among rational numbers with a universal-existential formula, American Journal of Mathematics, vol. 131 (2009), no. 3, pp. 675–682.Google Scholar
[23] Poonen, Bjorn, Elliptic curves whose rank does not grow and Hilbert's tenth problem over the rings of integers, private communication.Google Scholar
[24] Poonen, Bjorn and Shlapentokh, Alexandra, Diophantine definability of infinite discrete non-archimedean sets and diophantine models for large subrings of number fields, Journal für die Reine und Angewandte Mathematik, vol. 2005 (2005), pp. 27–48.Google Scholar
[25] Robinson, Julia, Definability and decision problems in arithmetic, The Journal of Symbolic Logic, vol. 14 (1949), pp. 98–114.Google Scholar
[26] Rogers, Hartley, Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[27] Shlapentokh, Alexandra, Extension of Hilbert's tenth problem to some algebraic number fields, Communications on Pure and Applied Mathematics, vol. XLII (1989), pp. 939–962.Google Scholar
[28] Shlapentokh, Alexandra, Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator, Inventiones Mathematicae, vol. 129 (1997), pp. 489–507.Google Scholar
[29] Shlapentokh, Alexandra, Defining integrality at prime sets of high density in number fields, Duke Mathematical Journal, vol. 101 (2000), no. 1, pp. 117–134.CrossRefGoogle Scholar
[30] Shlapentokh, Alexandra, Hilbert's tenth problem over number fields, a survey, Hilbert's tenth problem: Relations with arithmetic and algebraic geometry (Denef, Jan, Lipshitz, Leonard, Pheidas, Thanases, and Van Geel, Jan, editors), Contemporary Mathematics, vol. 270, American Mathematical Society, 2000, pp. 107–137.Google Scholar
[31] Shlapentokh, Alexandra, 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, vol. 95 (2002), pp. 227–252.Google Scholar
[32] Shlapentokh, Alexandra, Hilbert's tenth problem: Diophantine classes and extensions to global fields, Cambridge University Press, 2006.Google Scholar
[33] Shlapentokh, Alexandra, Elliptic curves retaining their rank in finite extensions and Hilbert's tenth problem for rings of algebraic numbers, Transactions of the American Mathematical Society, vol. 360 (2008), no. 7, pp. 3541–3555.CrossRefGoogle Scholar
[34] Shlapentokh, Alexandra, Rings of algebraic numbers in infinite extensions of ℚ and elliptic curves retaining their rank, Archive for Mathematical Logic, vol. 48 (2009), no. 1, pp. 77–114.CrossRefGoogle Scholar
[35] Shlapentokh, Alexandra, Using indices of points on an elliptic curve to construct a diophantine model of ℤ and define ℤ using one universal quantifier in very large subrings of number fields, including ℚ, arXiv:090l.4168v1 [math.NT].Google Scholar
[36] Silverman, Joseph, The arithmetic of elliptic curves, Springer Verlag, New York, New York, 1986.Google Scholar