Common extensions of valuations to rational function fields

Abstract

Let (K(X)|K,w) be a valuation transcendental extension of rational function fields and take a minimal pair of definition (a, gamma). In this paper, we characterize those K-conjugates a' of a such that the monomial valuation induced by the pair (a', gamma) restricts to w on K(X). In particular, we show that a' satisfies this if and only if a' and a are conjugates over the henselization of (K,v). The second part of the paper concerns abstract key polynomials. We introduce the notion of a regular limit key polynomial, and more generally, that of a regular complete sequence of key polynomials. We prove that regularity is equivalent to the property that every root of every key polynomial determines the corresponding truncated valuation. This extends earlier work of Mahboub, Mansour and Spivakovsky by allowing both limit key polynomials and valuation algebraic extensions. As a consequence, we obtain that w always admits a regular complete sequence of key polynomials whenever (K,v) is dense in its henselization.