Multiplicative valued difference fields (Q2892675)

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.