Discussion: On the Henselian Rationality Problem

The problem of local uniformization is one of the central problems in algebraic geometry and valuation theory. Closely connected is the problem of henselian rationality, which asks whether an immediate extension of valued function fields $(F|K,w)$ of transcendence degree one becomes rational after henselization, that is, if there exists some $X$ in the henselization $F^h$ of $F$ such that $F^h = K(X)^h$. Over tame fields, the problem was answered in the affirmative by Kuhlmann. However, the problem remains open if we remove the tameness hypothesis. In this talk, we will discuss some recent developments over perfect fields. We will first explain the key ideas behind Kuhlmann’s proof over tame fields and the obstacles that arise when one attempts to extend these ideas to the perfect-field setting. We will then discuss alternative approaches toward henselian rationality over perfect fields.