Extensions of Hilbert’s Tenth Problem: Definability and Decidability in Number Theory

DOI10.1007/978-3-319-41842-1_3zbMath1439.03081OpenAlexW2585631816MaRDI QIDQ3305316

Publication date: 6 August 2020

Published in: Outstanding Contributions to Logic (Search for Journal in Brave)

Full work available at URL: https://doi.org/10.1007/978-3-319-41842-1_3

zbMATH Keywords

Diophantine definability recursively enumerable sets Hilbert's tenth problem existential definability

Mathematics Subject Classification ID

Decidability (number-theoretic aspects) (11U05) Undecidability and degrees of sets of sentences (03D35) Decidability of theories and sets of sentences (03B25) Interpolation, preservation, definability (03C40)

Cites Work

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Additional results to a theorem of Eisenträger and Everest
Diophantine definability and decidability in large subrings of totally real number fields and their totally complex extensions of degree 2
Division-ample sets and the Diophantine problem for rings of integers
Ranks of twists of elliptic curves and Hilbert's tenth problem
Rings of algebraic numbers in infinite extensions of $\mathbb Q$ and elliptic curves retaining their rank
On $L$-functions of elliptic curves and cyclotomic towers
Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of $\mathbb{Q}$
Diophantine classes of holomorphy rings of global fields
Diophantine definability over some rings of algebraic numbers with infinite number of primes allowed in the denominator
Definability of the ring of integers in pro-$p$ Galois extensions of number fields
First-order decidability and definability of integers in infinite algebraic extensions of the rational numbers
Defining integrality at prime sets of high density in number fields.
Diophantine definability and decidability in extensions of degree 2 of totally real fields
Hilbert's tenth problem and Mazur's conjectures in complementary subrings of number fields
A universal first-order formula defining the ring of integers in a number field
ELLIPTIC CURVE POINTS AND DIOPHANTINE MODELS OF ℤ IN LARGE SUBRINGS OF NUMBER FIELDS
Definability of the ring of integers in some infinite algebraic extensions of the rationals
Decidability of the rings of real algebraic and p-adic algebraic integers.
TWO RECURSIVELY ENUMERABLE SETS OF INCOMPARABLE DEGREES OF UNSOLVABILITY (SOLUTION OF POST'S PROBLEM, 1944)
The Undecidability of Algebraic Rings and Fields
Extension of Hilbert's tenth problem to some algebraic number fields
Hilbert's Tenth Problem for a Class of Rings of Algebraic Integers
Elliptic curves retaining their rank in finite extensions and Hilbert's Tenth Problem for rings of algebraic numbers
Descent on elliptic curves and Hilbert’s tenth problem
Characterizing integers among rational numbers with a universal-existential formula
Arithmetic over the rings of all algebraic integers.
Elimination theory for the ring of algebraic integers.
Diophantine Sets Over Algebraic Integer Rings. II
Undecidability and Definability for the Theory of Global Fields
Hilbert's Tenth Problem for Quadratic Rings
Diophantine Sets over Some Rings of Algebraic Integers
On the constructible numbers
Questions of decidability and undecidability in Number Theory
Diophantine equivalence and countable rings
Hilbert's Tenth Problem is Unsolvable
Hilbert’s Tenth Problem and Mazur’s Conjecture for large subrings of $\mathbb {Q}$
Hilbert’s Tenth Problem: Relations with Arithmetic and Algebraic Geometry
Defining transcendentals in function fields
Model Theory
DECIDABLE ALGEBRAIC FIELDS
As easy as $\mathbb {Q}$: Hilbert’s Tenth Problem for subrings of the rationals and number fields
THE UNDECIDABILITY OF PURE TRANSCENDENTAL EXTENSIONS OF REAL FIELDS
Diophantine definability of infinite discrete nonarchimedean sets and Diophantine models over large subrings of number fields
Definability and decision problems in arithmetic

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