Abstract
In 1900 David Hilbert presented a list of questions at an international meeting of Mathematicians in Paris.
A. Shlapentokh—The author has been partially supported by the NSF grant DMS-1161456.
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References
Cornelissen, G., Thanases, P., Zahidi, K., Zahidi, K.: Division-ample sets and diophantine problem for rings of integers. J. de Théorie des Nombres Bordeaux 17, 727–735 (2005)
Cornelissen, G., Zahidi, K.: Topology of diophantine sets: remarks on Mazur’s conjectures. In: Denef, J., Lipshitz, L., Pheidas, T., Van Geel, J. (eds.) Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry. Contemporary Mathematics, pp. 253–260. American Mathematical Society, Rhods Island (2000)
Davis, M.: Hilbert’s tenth problem is unsolvable. Am. Math. Monthly 80, 233–269 (1973)
Davis, M., Matiyasevich, Y., Robinson, J.: Hilbert’s tenth problem. Diophantine equations: Positive aspects of a negative solution. In: Proceedings of Symposia in Pure Mathematics, vol. 28, pp. 323–378. American Mathematical Society (1976)
Denef, J.: Hilbert’s tenth problem for quadratic rings. Proc. Am. Math. Soc. 48, 214–220 (1975)
Denef, J., Lipshitz, L.: Diophantine sets over some rings of algebraic integers. J. London Math. Soc. 18(2), 385–391 (1978)
Eisenträger, K., Everest, G.: Descent on elliptic curves and Hilbert’s tenth problem. Proc. Am. Math. Soc. 137(6), 1951–1959 (2009)
Eisenträger, K., Everest, G., Shlapentokh, A.: Hilbert’s tenth problem and Mazur’s conjectures in complementary subrings of number fields. Math. Res. Lett. 18(6), 1141–1162 (2011)
Eisenträger, K., Miller, R., Park, J., Shlapentokh, A.: Easy as \(\mathbb{Q}\). (work in progress)
Friedberg, R.M.: Two recursively enumerable sets of incomparable degrees of unsolvability. Proc. National Acad. Sci. USA 43, 236–238 (1957)
Jochen K.: Defining \({\mathbb{Z}}\) in \({\mathbb{Q}}\) Annals of Mathematics to appear
Matiyasevich, Y.V.: Hilbert’s tenth problem. Foundations of Computing Series. MIT Press, Cambridge (1993). Translated from the 1993 Russian original by the author, With a foreword by Martin Davis
Mazur, B.: The topology of rational points. Exp. Math. 1(1), 35–45 (1992)
Mazur, B.: Questions of decidability and undecidability in number theory. J. Symb.Logic 59(2), 353–371 (1994)
Mazur, B.: Galois representations in arithmetic algebraic geometry. In: Scholl, A.J., Taylor, R.L. (eds.) Open problems regarding rational points on curves and varieties. Cambridge University Press, Cambridge (1998)
Mazur, B., Rubin, K.: Ranks of twists of elliptic curves and Hilbert’s Tenth Problem. Inventiones Mathematicae 181, 541–575 (2010)
Muchnik, A.A.: On the separability of recursively enumerable sets. Doklady Akademii Nauk SSSR (N.S.), vol. 109,pp. 29–32 (1956)
Perlega, S.: Additional results to a theorem of Eisenträger and Everest. Archiv der Mathematik (Basel) 97(2), 141–149 (2011)
Pheidas, T.: Hilbert’s tenth problem for a class of rings of algebraic integers. Proc. Am. Math. Soc. 104(2), 611–620 (1988)
Bjorn, P.: Elliptic curves whose rank does not grow and Hilbert’s Tenth Problem over the rings of integers. Private Communication
Poonen, B.: Using elliptic curves of rank one towards the undecidability of Hilbert’s tenth problem over rings of algebraic integers. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 33–42. Springer, Heidelberg (2002)
Poonen, B.: Hilbert’s tenth problem and Mazur’s conjecture for large subrings of \(\mathbb{Q}\). J Am. Math. Soc. 16(4), 981–990 (2003)
Poonen, B., Shlapentokh, A.: Diophantine definability of infinite discrete non-archimedean sets and diophantine models for large subrings of number fields. Journal für die Reine und Angewandte Mathematik 27–48, 2005 (2005)
Robinson, J.: Definability and decision problems in arithmetic. J. Symb. Logic 14, 98–114 (1949)
Shlapentokh, A.: Extension of Hilbert’s tenth problem to some algebraic number fields. Commun. Pure Appl. Math. XLII, 939–962 (1989)
Shlapentokh, A.: Diophantine classes of holomorphy rings of global fields. J. Algebra 169(1), 139–175 (1994)
Shlapentokh, A.: Diophantine equivalence andcountable rings. J. Symb. Logic 59, 1068–1095 (1994)
Shlapentokh, A.: Elliptic curves retaining their rank in finite extensions and Hilbert’s tenth problem for rings of algebraic numbers. Trans. Am. Math. Soc. 360(7), 3541–3555 (2008)
Shlapentokh, A.: Elliptic curve points and Diophantine models of \(\mathbb{Z}\) in large subrings of number fields. Int. J. Number Theor. 8(6), 1335–1365 (2012)
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Shlapentokh, A. (2015). Hilbert’s Tenth Problem for Subrings of \({\mathbb {Q}}\) and Number Fields (Extended Abstract). In: Jain, R., Jain, S., Stephan, F. (eds) Theory and Applications of Models of Computation. TAMC 2015. Lecture Notes in Computer Science(), vol 9076. Springer, Cham. https://doi.org/10.1007/978-3-319-17142-5_1
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