Diophantine undecidability of \({\mathbb{C}{}}(t_ 1,t_ 2)\)
type: Item
instance of: scholarly article
title: Diophantine undecidability of \({\mathbb{C}{}}(t_ 1,t_ 2)\)
zbMATH Open document ID: 0754.11039
DOI: 10.1016/S0021-8693(05)80047-X
published in: Journal of Algebra
publication date: 1992-09-27T00:00:00Z
review text: The main result of this paper is that the Diophantine theory of the complex function field \(\mathbb{C}(t_ 1,t_ 2)\), where \(t_ 1\) are algebraically independent over the complex numbers, in the language \(\{+,\times,t_ 1,t_ 2,0,1\}\) is undecidable. A model of \((\mathbb{Z},+,\times)\) represented by isogenies of an elliptic curve is also constructed.
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 11D99
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 63892
zbMATH Keywords: undecidability
zbMATH Keywords: Diophantine theory
zbMATH Keywords: complex function field
author: Ki Hang Kim
author: Fred Roush
reviewed by: Dimitru Buşneag
MaRDI profile type: Publication
cites work: Remarks on the elementary theories of formal and convergent power series
cites work: The Diophantine Problem for Polynomial Rings and Fields of Rational Functions
cites work: Q5666543
cites work: Q4143433
cites work: Problems equivalent to rational Diophantine solvability
cites work: Q5675788
cites work: Units and class groups in number theory and algebraic geometry
cites work: Q3315578
cites work: Q3710637
cites work: On the decidability of Diophantine problems in combinatorial geometry
cites work: Q5723265
Recommended article: Undecidability of ℚ⁽²⁾
Recommended article: Diophantine undecidability in some rings of algebraic numbers of totally real infinite extensions of \(\mathbb{Q}\)
Recommended article: Diophantine undecidability for some function fields of infinite transcendence degree and positive characteristic
Recommended article: Undecidability in diagonalizable algebras
Recommended article: Diophantine undecidability over algebraic function fields over finite fields of constants
Recommended article: Diophantine undecidability of holomorphy rings of function fields of characteristic 0
Recommended article: Diophantine undecidability for some holomorphy rings of algebraic functions of characteristic 0.
Recommended article: Q5567866
Recommended article: The undecidability of cyclotomic towers
Recommended article: Q4546151
description: scientific article; zbMATH DE number 63892