Hilbert's tenth problem for function fields of varieties over algebraically closed fields of positive characteristic
type: Item
instance of: scholarly article
title: Hilbert's tenth problem for function fields of varieties over algebraically closed fields of positive characteristic
zbMATH Open document ID: 1270.11126
author: Kirsten Eisenträger
published in: Monatshefte für Mathematik
publication date: 2012-11-08T00:00:00Z
review text: Let \(K\) be a function field of a variety of dimension \(\geq 2\) over an algebraically closed field \(k\) of characteristic \(p > 2\). It is proved that there exist elements \(z_1, z_2 \in K\) which are algebraically independent over \(k\) such that Hilbert's Tenth Problem for \(K\) with coefficients in \({\mathbb F}_p[z_1, z_2]\) is undecidable. The proof works by interpreting the ring of endomorphisms of an elliptic curve with division and two special pair relations in the given structure, by existential formulas. Endomorphisms are related to points on twists, and good twists are chosen using a theorem by Moret-Bailly. The result is both a generalisation and a week form of a result by Kim and Roush; week form because by Kim and Roush the two algebaric independent elements were canonic.
reviewed by: Mihai Prunescu
Mathematics Subject Classification ID: 11U05
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 6103716
zbMATH Keywords: Hilbert's tenth problem
zbMATH Keywords: undecidability
zbMATH Keywords: elliptic curves
zbMATH Keywords: function fields
MaRDI profile type: Publication
full work available at URL: s00605-011-0364-7
OpenAlex ID: W2004744288
cites work: Q4657443
cites work: Hilbert's Tenth Problem for Quadratic Rings
cites work: The Diophantine Problem for Polynomial Rings and Fields of Rational Functions
cites work: Lectures on the theory of algebraic functions of one variable
cites work: Hilbert's tenth problem for algebraic function fields of characteristic 2.
cites work: Q4658943
cites work: Hilbert's Tenth problem for function fields of varieties over number fields and \(p\)-adic fields
cites work: Diophantine undecidability of \({\mathbb{C}{}}(t_ 1,t_ 2)\)
cites work: Diophantine unsolvability for function fields over certain infinite fields of characteristic \(p\)
cites work: Diophantine unsolvability over \(p\)-adic function fields
cites work: Q5287502
cites work: Elliptic curves and Hilbert’s tenth problem for algebraic function fields over real andp-adic fields
cites work: Q4850044
cites work: Hilbert's tenth problem for fields of rational functions over finite fields
cites work: Undecidability of existential theories of rings and fields: A survey
cites work: Diophantine undecidability over algebraic function fields over finite fields of constants
cites work: Hilbert's tenth problem for algebraic function fields over infinite fields of constants of positive characteristic
cites work: Q4546151
cites work: Q3710637
cites work: Hilbert's Tenth Problem for Rational Function Fields in Characteristic 2
DOI: 10.1007/S00605-011-0364-7
description: scientific article; zbMATH DE number 6103716