Difference equations in Witt vectors
type: Item
instance of: scholarly article
title: Difference equations in Witt vectors
zbMATH Open document ID: 1107.13029
author: Luc Bélair
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2005-02-23T00:00:00Z
review text: Let \(A\) be a discrete valuation ring of characteristic \(0\) with uniformizing parameter \(t\) and let \(\sigma\) be an automorphism of \(A\). Let \(A[X]_{\sigma}\) denote the ring of polynomials in the indeterminates \(\sigma^i(X)\) for \(i\in{\mathbb Z}\). Assume that \(A\) satisfies an analogue of Hensel's lemma for polynomials in \(A[X]_{\sigma}\). The author sketches a proof of the following analogue of a theorem of \textit{M. J. Greenberg} [Publ. Math., Inst. Hautes Étud. Sci. 31, 563--568 (1966; Zbl 0146.42201)]: given \(f_{1}\), \dots, \(f_{m}\in A[X_{1},\dots,X_{n}]_{\sigma}\), there exists an integer \(N\) such that for every integer \(\alpha>0\) and every \(x_{1}\), \dots, \(x_{n}\in A\) such that \(f_{1}(x_{1},\dots,x_{n})\equiv\dots\equiv f_{m}(x_{1},\dots,x_{n})\equiv0 \mod t^{\alpha N}\), there exist \(y_{1}\), \dots, \(y_{n}\in A\) such that \(f_{1}(y_{1},\dots,y_{n})=\dots=f_{m}(y_{1},\dots,y_{n})=0\) and \(y_{i}\equiv x_{i}\mod t^\alpha\) for \(i=1\), \dots, \(n\). This result is applied to obtain a Nullstellensatz for the field of fractions of the ring of Witt vectors over the algebraic closure of the field of \(p\) elements, with \(\sigma\) the Frobenius automorphism. Detailed proofs will appear elsewhere.
reviewed by: Jean-Pierre Tignol
Mathematics Subject Classification ID: 13K05
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 2137994
MaRDI profile type: Publication
full work available at URL: j.crma.2004.12.001
OpenAlex ID: W2090645465
cites work: Ultraproducts and approximation in local rings. I
cites work: Integer-valued difference rational functions over Witt vectors
cites work: L'automorphisme de Frobenius des vecteurs de Witt
cites work: An approximation property for Teichmüller points
cites work: Q5334359
cites work: Q3331307
cites work: Rational points in Henselian discrete valuation rings
cites work: Strictly local solutions of diophantine equations
cites work: Q5654135
cites work: Q4081371
cites work: Model Theory
cites work: Q4510745
cites work: Elementary embeddings of fields of power series
cites work: Q4440912
DOI: 10.1016/J.CRMA.2004.12.001
Recommended article: Q3754086
Recommended article: Approximation for Frobenius algebraic equations in Witt vectors
Recommended article: Planes of the form \(b(X,Y)Z^n-a(X,Y)\) over a DVR
Recommended article: Q3491710
Recommended article: New light on Hensel's Lemma
Recommended article: Q4295697
Recommended article: Q3834152
Recommended article: Split primes and integer-valued polynomials
Recommended article: A general version of the Nullstellensatz for arbitrary fields
Recommended article: Hilbert's Tenth Problem for Rings of Algebraic Functions in One Variable Over Fields of Constants of Positive Characteristic
description: scientific article; zbMATH DE number 2137994