Multiplicative valued difference fields
type: Item
instance of: scholarly article
title: Multiplicative valued difference fields
zbMATH Open document ID: 1245.03056
author: Koushik Pal
publication date: 2012-06-19T00:00:00Z
full work available at URL: https://arxiv.org/abs/1011.1655
full work available at URL: https://projecteuclid.org/euclid.jsl/1333566637
Mathematics Subject Classification ID: 03C60
Mathematics Subject Classification ID: 03C10
Mathematics Subject Classification ID: 12H10
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12L12
zbMATH DE Number: 6047776
zbMATH Keywords: difference fields
zbMATH Keywords: valued fields
zbMATH Keywords: RV-structure
zbMATH Keywords: quantifier elimination
MaRDI profile type: Publication
OpenAlex ID: W2964171304
arXiv ID: 1011.1655
cites work: Model theory of the Frobenius on the Witt vectors
cites work: Elementary theory of valued fields with a valuation-preserving automorphism
cites work: Valued fields with contractive automorphism and Kaplansky fields
cites work: Existentially closed ordered difference fields and rings
cites work: Model Theory
cites work: An isomorphism theorem for Henselian algebraic extensions of valued fields
DOI: 10.2178/JSL/1333566637
published in: The Journal of Symbolic Logic
review text: The valued difference fields of the title are valued fields \((K,v)\) with a distinguished automorphism \(\sigma\) inducing an automorphism of the valuation ring. Such an automorphism readily induces an automorphism of the residue field, and an order-preserving automorphism of the valued group. The basic model theory of these structures has been mostly worked out in case the automorphism is an isometry, viz. \(v(\sigma(x))=v(x)\), or is such that \(v(\sigma(x))> v(x^n)\) for all \(n\in \mathbb N\) when \(v(x)>0\), generalizing the classical work of Ax-Kochen and Ershov, with the usual proviso on characteristics.NEWLINENEWLINEIn this paper, the author generalizes the above previous results in equicharacteristic zero by studying the case where the induced automorphism of the value group is some fixed scalar multiplication when the value group is viewed as a module over an appropriate real closed field, thus \(v(\sigma(x))=\rho \cdot v(x)\), for some fixed \(\rho\) in the real closed field. He gives an axiomatization, and a quantifier elimination relative to the ``residue valuative structure'', RV-structure, of the quotient group \(K^\times/1+\mathfrak m\), viz. the multiplicative group of \(K\) modulo the units of the valuation ring of the form \(1+x\) with \(x\) an element of the maximal ideal. Note in passing that, muddling conventional valued fields terminology, the ``cross-sections'' of the author (of the valuation map) go from the value group into \(K^\times/1+\mathfrak m\), rather than into \(K^\times\), and thus are the ``other side'' of the usual angular component maps corresponding to a splitting of the RV-structure with respect to the induced valuation map.
reviewed by: Luc Bélair
description: scientific article; zbMATH DE number 6047776