Diophantine undecidability over algebraic function fields over finite fields of constants
type: Item
instance of: scholarly article
title: Diophantine undecidability over algebraic function fields over finite fields of constants
zbMATH Open document ID: 0856.11058
author: Alexandra Shlapentokh
published in: Journal of Number Theory
publication date: 1997-03-05T00:00:00Z
review text: Let \(F(t)\) be a function field over a finite field \(F\). Let \(K\) be a finite extension of \(F(t)\). One can ask the following analogue of Hilbert's Tenth problem for \(K\): Is the existential theory of \(F(t)\) in the language \(\{0,1,t,+,\cdot\}\) decidable, that is, is there an algorithm which checks the existence of solutions \((x_1,\dots,x_n)\) of an arbitrary polynomial equation \(f=0\) in \(K\) with \(f\in F[t,X_1,\dots, X_n]\)? The author shows that the answer to this question is ``No''. This is the first result for algebraic function fields of arbitrary degree over \(F(t)\) and may be an evidence for the similar question for number fields, an outstanding open problem. The similar question for arbitrary algebraic function fields over any \(F\) remains open.
reviewed by: Thanases Pheidas
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 03B25
Mathematics Subject Classification ID: 11N99
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 894610
zbMATH Keywords: diophantine undecidability
zbMATH Keywords: function field over a finite field
zbMATH Keywords: analogue of Hilbert's Tenth problem
zbMATH Keywords: existential theory
zbMATH Keywords: existence of solutions
MaRDI profile type: Publication
full work available at URL: jnth.1996.0080
OpenAlex ID: W2035776188
DOI: 10.1006/JNTH.1996.0080
description: scientific article; zbMATH DE number 894610