Hilbert's Tenth Problem for Subrings of ℚ and Number Fields (Extended Abstract)

In 1900 David Hilbert presented a list of questions at an international meeting of Mathematicians in Paris.

[1]  Topology of diophantine sets: remarks on Mazur's conjectures , 2000, math/0006140.

[2]  Alexandra Shlapentokh,et al.  Hilbert's Tenth Problem and Mazur's Conjectures in Complementary Subrings of Number Fields , 2010, 1012.4878.

[3]  Hilbert’s tenth problem for quadratic rings , 1975 .

[4]  ELLIPTIC CURVE POINTS AND DIOPHANTINE MODELS OF ℤ IN LARGE SUBRINGS OF NUMBER FIELDS , 2012 .

[5]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[6]  Jan Denef Hilbert's tenth problem : relations with arithmetic and algebraic geometry : Workshop on Hilbert's Tenth Problem : Relations with Arithmetic and Algebraic Geometry, November 2-5, 1999, Ghent University, Belgium , 2000 .

[7]  Martin D. Davis Hilbert's Tenth Problem is Unsolvable , 1973 .

[8]  J. Koenigsmann Defining $\mathbb{Z}$ in $\mathbb{Q}$ , 2010, 1011.3424.

[9]  Bjorn Poonen Using Elliptic Curves of Rank One towards the Undecidability of Hilbert's Tenth Problem over Rings of Algebraic Integers , 2002, ANTS.

[10]  Barry Mazur,et al.  The Topology of Rational Points , 1992, Exp. Math..

[11]  T. Pheidas,et al.  Division-ample sets and the Diophantine problem for rings of integers , 2003, math/0312382.

[12]  K. Rubin,et al.  Ranks of twists of elliptic curves and Hilbert’s tenth problem , 2009, 0904.3709.

[13]  Alexandra Shlapentokh Diophantine Classes of Holomorphy Rings of Global Fields , 1994 .

[14]  Alexandra Shlapentokh Extension of Hilbert's tenth problem to some algebraic number fields , 1989 .

[15]  B. Poonen Hilbert's Tenth Problem and Mazur's Conjecture for large subrings of $\mathbb{Q}$ , 2003, math/0306277.

[16]  B. Poonen,et al.  Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields , 2004, math/0408271.

[17]  L. Lipshitz,et al.  Diophantine Sets over Some Rings of Algebraic Integers , 1978 .

[19]  Alexandra Shlapentokh Elliptic curves retaining their rank in finite extensions and Hilbert’s Tenth Problem for rings of algebraic numbers , 2008 .

[20]  Barry Mazur,et al.  Questions of decidability and undecidability in Number Theory , 1994, Journal of Symbolic Logic.

[21]  Descent on elliptic curves and Hilbert’s tenth problem , 2007, 0707.1485.

[22]  Yuri Matiyasevich,et al.  Hilbert’s tenth problem , 2019, 100 Years of Math Milestones.

[23]  Julia Robinson,et al.  Definability and decision problems in arithmetic , 1949, Journal of Symbolic Logic.

[24]  T. Pheidas Hilbert’s tenth problem for a class of rings of algebraic integers , 1988 .

[25]  R. Friedberg,et al.  TWO RECURSIVELY ENUMERABLE SETS OF INCOMPARABLE DEGREES OF UNSOLVABILITY (SOLUTION OF POST'S PROBLEM, 1944). , 1957, Proceedings of the National Academy of Sciences of the United States of America.