Absorbent property, Krasner type lemmas and spectral norms for a class of valued fields (Q2448589)

Let \((K,\varphi)\) be a perfect valued field of rank \(1\). For any extension \(\bar{\varphi}\) of \(\varphi\) to a fixed algebraic closure \(\overline{K}\) of \(K\), we define the \textit{\(\varphi\)-spectral norm} of \(\overline K\) by \(\|x\|_{\varphi} :=\max\{\bar{\varphi}(\sigma(x)): \sigma\in G\}\), where \(G:=\mathrm{Gal}(\overline{K}/K)\). We have that \(\|\cdot\|_{\varphi}\) does not depend on \(\bar{\varphi}\). We say that \(\bar{\varphi}\) is \textit{\(G\)-equivariant} if \(\bar{\varphi}\circ \sigma=\bar{\varphi}\) for all \(\sigma \in G\). The main result of the paper is the following ``Absorbent Theorem'': Let \(L\) be a subfield, \(K\subseteq L \subseteq \overline{K}\) and let \(\omega(\alpha):=\min\limits_{ \sigma\in G}\{\|\alpha-\sigma( \alpha)\|_{\varphi} : \alpha\neq \sigma(\alpha)\}\) if \(\alpha\notin K\) and \(\omega(\alpha)=0\) if \(\alpha\in K\). Let \(\alpha\in \bar{K}\setminus K\) be such that \(\inf\limits_{\beta\in L} \|\alpha-\beta\|_{\varphi} < c_{\ast} \omega (\alpha)\), where \(c_{\ast}=1/2\) if \(\varphi\) is Archimedean and \(c_{\ast} =1\) if \(\varphi\) is non-Archimedean. Then \(\alpha\in L\). The same is true for any \(G\)-equivariant \(\varphi\)-norm on \(\overline{K}\). As a consequence it is obtained the following generalization of Krasner's Lemma: Let \(\alpha\in \overline{K}\setminus K\) and let \(y\in \overline{K}\) be such that \(\|\alpha-y\|_{\varphi}<c_{\ast}\omega(\alpha)\). Then \(K(\alpha) \subseteq K(y)\). Next, the author generalizes the absorbent theorem to a class of closed subrings \(L\subseteq \widetilde{\overline{K}}\) which are not necessarily algebraic over \(K\). Here \(\widetilde{\overline{K}}\) is the completion of \(\overline{K}\) with respect to the \(\varphi\)-spectral norm (\(\widetilde{\overline{K}}\) is a field if and only if \(\|\cdot\|_{\varphi}\) is a multiplicative absolute value). Also it is established a Krasner's Lemma for \(\widetilde{\overline{K}}\). Finally, using these results, the author discusses in a more general context the following conjecture of A. Zaharescu: For any \(x,y\in {\mathbb C}_p\), the complex \(p\)--adic field, there exists \(t\in{\mathbb Q}_p\) such that \(\widetilde{\mathbb Q}_p(x,y)=\widetilde{\mathbb Q}_p(x+ty)\) where \(\tilde{L}\) means the \(p\)--adic topological closure of a subfield \(L\) of \(\mathbb C_p\) in \(\mathbb C_p\).