Metric invariants over Henselian valued fields. (Q1403858)

Let \(v\) be a Henselian valuation of a field \(K\) and \(\overline{v}\) be the prolongation of \(v\) to a fixed algebraic closure \(\overline{K}\) of \(K\). In this paper the authors introduce the notion of an \(\varepsilon\)-generalized distinguished sequence associated to any separable element of \(\overline{K} \setminus K\) and to an infinitesimal quantity belonging to the value group of \(\overline{v}\). These \(\varepsilon\)-generalized distinguished sequences which are generalizations of saturated distinguished sequences [cf. \textit{N. Popescu} and \textit{A. Zaharescu}, J. Number Theory 52, 98--118 (1995; Zbl 0838.11078); \textit{K. Ota}, J. Number Theory 79, 217--248 (1999; Zbl 0999.11071)] are used to study various arithmetic and metric invariants of elements of \(\overline{K}{}^{\text{sep}} \setminus K\). In the process, some Ax-Sen type inequalities have also been obtained [see \textit{J. Ax}, J. Algebra 15, 417--428 (1970; Zbl 0216.04703)].