Division-ample sets and the Diophantine problem for rings of integers

DOI10.5802/jtnb.516zbMath1161.11323arXivmath/0312382OpenAlexW1992184757MaRDI QIDQ819874

Gunther Cornelissen, Thanases Pheidas, Zahidi, Karim

Publication date: 30 March 2006

Published in: Journal de Théorie des Nombres de Bordeaux (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/math/0312382

Mathematics Subject Classification ID

Decidability (number-theoretic aspects) (11U05) Diophantine equations in many variables (11D72) Algebraic numbers; rings of algebraic integers (11R04)

Related Items (15)

Diophantine definability and decidability in extensions of degree 2 of totally real fields ⋮ ELLIPTIC CURVE POINTS AND DIOPHANTINE MODELS OF ℤ IN LARGE SUBRINGS OF NUMBER FIELDS ⋮ NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES ⋮ First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers ⋮ Existential decidability for addition and divisibility in holomorphy subrings of global fields ⋮ Elliptic curves, \(L\)-functions, and Hilbert's tenth problem ⋮ Existential definability and diophantine stability ⋮ Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers ⋮ Superficies elípticas y el décimo problema de Hilbert ⋮ As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields ⋮ Extensions of Hilbert’s Tenth Problem: Definability and Decidability in Number Theory ⋮ Defining Integers ⋮ Rings of algebraic numbers in infinite extensions of \(\mathbb Q\) and elliptic curves retaining their rank ⋮ Diophantine undecidability of holomorphy rings of function fields of characteristic 0 ⋮ Towards Hilbert's tenth problem for rings of integers through Iwasawa theory and Heegner points

Cites Work

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• Extension of Hilbert's tenth problem to some algebraic number fields
• Hilbert's Tenth Problem for a Class of Rings of Algebraic Integers
• Diophantine relationships between algebraic number fields
• Diophantine Sets Over Algebraic Integer Rings. II
• Hilbert's Tenth Problem for Quadratic Rings
• Diophantine Sets over Some Rings of Algebraic Integers
• The orders of the reductions of a point in the Mordell-Weil group of an elliptic curve
• Computing the Rank of Elliptic Curves over Number Fields

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