On the implicit constant fields and key polynomials for valuation algebraic extensions

Extensions of valuations to rational function fields over completions

“…As a consequence, $$\lbrace a_\nu \rbrace _{\nu <\lambda }$$ is a pCs of transcendental type in $$(\overline{\widehat{K}}, \overline{\widehat{v}})$$ if and only if the same holds for any other pair of sequences $$\lbrace \gamma ^\prime _{\nu ^\prime }\rbrace _{\nu ^\prime <\lambda ^\prime } \subseteq \overline{w}(X-\overline{K})$$ and $$\lbrace a^\prime _{\nu ^\prime }\rbrace _{\nu ^\prime < \lambda ^\prime } \subseteq \overline{K}$$ satisfying the conditions ( C 1) and ( C 2). Applying the construction in [8, Section 5], without any loss of generality, we can now assume that $$\lbrace a_\nu \rbrace _{\nu <\lambda }$$ and $$\lbrace \gamma _\nu \rbrace _{\nu <\lambda }$$ satisfy the following additional condition: $$$$\begin{equation} (a_\nu ,\gamma _\nu ) \text{ is a minimal pair of definition for $\overline{v}_{a_\nu ,\gamma _\nu }$ over }K. \end{equation}$$$$Observe that …”

“…Proof We have observed in [8, section 5] that we can construct sequences $$\lbrace \gamma _\nu \rbrace _{\nu <\lambda } \subseteq v(X-\overline{K})$$ and $$\lbrace a_\nu \rbrace _{\nu <\lambda } \subseteq \overline{K}$$ such that $$\lbrace \gamma _\nu \rbrace _{\nu <\lambda }$$ is cofinal in $$v(X-\overline{K})$$, $$v(X-a_\nu ) = \gamma _\nu$$ and $$(a_\nu ,\gamma _\nu )$$ is a minimal pair of definition for $$v_{a_\nu ,\gamma _\nu }$$ over K for all $$\nu <\lambda$$. $$(K(X)|K,v)$$ being a valuation algebraic extension implies that $$(\overline{K}(X)|\overline{K},v)$$ is immediate and hence $$\gamma _\nu \in v\overline{K}$$.…”

“…The assumption that $$K^h$$ is separably tame implies that $$K^\mathrm{sep}= K^r$$. It now follows from [8, Theorem 6.2] that $$$$\begin{equation*} IC(K(X)|K,v) = K^h(a_\nu \mid \nu <\lambda ). \end{equation*}$$$$It follows from Theorem 4.2 that $$\lbrace a_\nu \rbrace _{\nu <\lambda }$$ is a Cauchy sequence in $$(\overline{K},v)$$ and the unique limit $$a\in \widehat{\overline{K}}$$ is algebraic over $$\widehat{K}$$.…”

“…A thorough understanding of this extension problem has given rise to a whole host of literature and several different interconnected notions. The fundamental objects of study are minimal pairs of definition [1][2][3][4], key polynomials [16][17][18]20], pseudo-Cauchy sequences (pCs) [9,19], and implicit constant fields [6][7][8]13]. We refer the reader to Section 2 for all the relevant definitions.…”