Multiplicative Valued Difference Fields

TL;DR

This paper introduces and analyzes the theory of multiplicative valued difference fields, extending previous work on isometric and contractive cases by providing axioms and a quantifier elimination result.

Contribution

It develops a new framework for multiplicative valued difference fields with a real-closed scalar, including axiomatization and quantifier elimination.

Findings

Established axioms for multiplicative valued difference fields.

Proved a relative quantifier elimination theorem for the theory.

Extended the understanding of valuation automorphism interactions.

Abstract

The theory of valued difference fields $(K, \sigma, v)$ depends on how the valuation $v$ interacts with the automorphism $\sigma$. Two special cases have already been worked out - the isometric case, where $v(\sigma(x)) = v(x)$ for all $x\in K$, has been worked out by Luc Belair, Angus Macintyre and Thomas Scanlon; and the contractive case, where $v(\sigma(x)) > n\cdot v(x)$ for all $n\in\mathbb{N}$ and $x\in K^\times$ with $v(x) > 0$, has been worked out by Salih Azgin. In this paper we deal with a more general version, called the multiplicative case, where $v(\sigma(x)) = \rho\cdot v(x)$, where $\rho (> 0)$ is interpreted as an element of a real-closed field. We give an axiomatization and prove a relative quantifier elimination theorem for such a theory.