Completion of r.t. extensions of local fields. I
Let \((K,\nu)\) be a local field and \((K(X),w_0)\) the so called Gauss residual transcendental extension of it; so, if \(P(X)=a_0+a_1 X+\dots+a_k X^k\) is a polynomial in \(K[ X]\) we have \(w_0 (P(X))=\inf_{0\leq i\leq k} \{\nu (a_i)\}\). We denote by \(K\{\{ X\}\}\) the local field of all formal Laurent series \(\alpha=\sum_{n\in \mathbb{Z}} a_n X^n\) such that \(\{\nu (a_n) \}_n\) is lower bounded and \(\nu(a_{-n} )\to \infty\) if \(n\to \infty\), with the valuation \(u\) where \(u(\alpha)=\inf_{n\in \mathbb{Z}} \{\nu (a_n)\}\). We consider the embedding \(\omega\) of the valued field \((K (X), w_0)\) in the local field \((K\{\{ X\}\},u)\) such that \[ \omega (P(X))=P(X), \quad \text{for any polynomial} \quad P(X)\in K[ X]; \tag{1} \] if \(P(X)=\prod^s_{i=1} (X- a_i)\), \(a_i\in \overline {K}\), the algebraic closure of \(K\), then \[ \omega (1/ P(X))=\omega \Biggl( \prod^s_{i=1} 1/(X- a_i) \Biggr)=(1/X^s) \prod^s_{i=1} \sum^\infty_{n=0} (a_i/ X)^n \tag{2} \] if \(\nu (a_i )>0\), for any \(i=1,\dots, s\), and \[ \omega \Biggl( \prod^s_{i=1} 1/ (X- a_i) \Biggr)=\prod^s_{i=1} (-1/ a_i) \sum^\infty_{n=0} (X/ a_i)^n \quad \text{if } \nu (a_i) \leq 0, \quad \text{for any }i=1, \dots, s. \tag{3} \] In this way we look at \((\widetilde {K(X)}, \widetilde {\omega}_0)\) as a valued subfield of \((K\{\{ X\}\}, \mu)\) where \(\widetilde {K(X)}\) is the completion of \(K(X)\) with respect to \(\omega_0\). A series \(\alpha=\sum_{n\geq 0} a_n X^n\) is called almost periodic if it is a limit in \((K \{\{ X\}\}, u)\) of elements from \(K(X)\). For any series \(\alpha=\sum_{n\geq 0} a_n X^n\) and for any two natural numbers \(m\), \(l\) we denote by \(\widetilde {H}^l_m (\alpha)\) the set of all determinants of order \(l+1\) formed by the first \(l+1\) columns and any \(l+1\) rows of the Hankel matrix \(H_m (\alpha)\) [see for instance \textit{Y. Amice}, Les nombres \(p\)-adiques, Paris (1975; Zbl 0313.12104)]. The fundamental result in this paper is Theorem 2.3: Let \(\alpha=\sum_{i\geq 0} a_i X^i\in K\{\{ X\}\}\) be such that \(u(\alpha) =0\). The following assertions are equivalent: 1) \(\alpha\) is an almost periodic series; 2) For any natural number \(n\) there exist two natural numbers, \(d_n=d_n (\alpha)\), \(m_n=m_n (\alpha)\) and elements \(q_{d_n}^{(n)}, \dots, q_0^{(n)}\) in \(O_\nu\) such that \(q_0^{(n)}\) is a unit, and for any \(m\geq m_n\) one has: \(\nu (q_{d_n} ^{(n)} a_m+\dots+q_0^{(n)} a_{m+d_n})\geq n\); 3) For any natural number \(n\) there exist natural numbers \(l_0=l_0 (n, \alpha)\), \(m_0=m_0 (n, \alpha)\), such that for \(l\geq l_0\) and \(m\geq m_0\) one has \(\nu (D)\geq nl+1\) for any \(D\in \widetilde {H}^l_m (\alpha)\).
local field of all formal Laurent series
scientific article; zbMATH DE number 923403