The logic of pseudo-\(S\)-integers
type: Item
instance of: scholarly article
title: The logic of pseudo-\(S\)-integers
zbMATH Open document ID: 0892.11037
DOI: 10.1007/BF02760931
author: Alexandra Shlapentokh
published in: Israel Journal of Mathematics
publication date: 1998-08-03T00:00:00Z
review text: Let \(\{n_i\}\) be a sequence of natural numbers and let \(\{p_i\}\) be a listing of rational primes. Then an abelian group \(G=\{x\in \mathbb{Q}\mid \text{ord}_{p_i} \geq-n_i\}\) is called a group of pseudo-integers. The scope of this paper is to investigate the logical properties of such groups of pseudo-integers and the counterparts of such groups in global fields in the case the number of primes allowed to appear in the denominator is infinite. The author shows that, while the addition problem of any recursive group of pseudo-integers is decidable, the diophantine problem for some recursive groups of pseudo-integers with infinite number of primes allowed in the denominator, is not decidable. More precisely, there exist recursive groups of pseudo-integers, where an infinite number of primes are allowed to appear in the denominator, such that there is no uniform algorithm to decide whether a polynomial equation over \(\mathbb{Z}\) in several variables has solutions in the group (this result is obtained by giving a diophantine definition of \(\mathbb{Z}\) over these groups; the proof is based on the strong Hasse norm principle).
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 1123745
zbMATH Keywords: decidability
zbMATH Keywords: Galois extension
zbMATH Keywords: group of pseudo-integers
zbMATH Keywords: addition problem
zbMATH Keywords: diophantine problem
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
cites work: Hilbert's Tenth Problem is Unsolvable
cites work: Q4115143
cites work: Hilbert's Tenth Problem for Quadratic Rings
cites work: Diophantine Sets Over Z[ T ]
cites work: The Diophantine Problem for Polynomial Rings and Fields of Rational Functions
cites work: Diophantine Sets over Some Rings of Algebraic Integers
cites work: Q3905379
cites work: Diophantine Sets Over Algebraic Integer Rings. II
cites work: On groups of pseudo-integers
cites work: Q4063245
cites work: Diophantine unsolvability for function fields over certain infinite fields of characteristic \(p\)
cites work: Diophantine undecidability of \({\mathbb{C}{}}(t_ 1,t_ 2)\)
cites work: Diophantine unsolvability over \(p\)-adic function fields
cites work: Q5612629
cites work: Q4022705
cites work: Questions of decidability and undecidability in Number Theory
cites work: Q4838055
cites work: Q5675787
cites work: Hilbert's Tenth Problem for a Class of Rings of Algebraic Integers
cites work: Hilbert's tenth problem for fields of rational functions over finite fields
cites work: Q4324957
cites work: On extensions of pseudo-integers
cites work: Q4298769
cites work: Constructions in Algebra
cites work: Diophantine relationships between algebraic number fields
cites work: Extension of Hilbert's tenth problem to some algebraic number fields
cites work: Hilbert's Tenth Problem for rings of algebraic functions of characteristic 0
cites work: Hilbert's Tenth Problem for Rings of Algebraic Functions in One Variable Over Fields of Constants of Positive Characteristic
cites work: A Diophantine definition of rational integers over some rings of algebraic numbers
cites work: Diophantine classes of holomorphy rings of global fields
cites work: Diophantine undecidability for some holomorphy rings of algebraic functions of characteristic 0.
cites work: Hilbert's Tenth Problem for Rational Function Fields in Characteristic 2
cites work: Q4090435
description: scientific article; zbMATH DE number 1123745