Definability of henselian valuations by conditions on the value group

Abstract

Given a henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a $t$-henselian non-henselian ordered field elementarily equivalent to a henselian field with a specified value group.