Reaching fixed points as limits in subspaces
Let \(X\) be a non-empty set, and \(\Gamma\) a partially ordered set with the smallest element \(0\). A mapping \(d:X\times X\to\Gamma\) is said to be an ultrametric distance if it satisfies the following three conditions for all \(x,y,z\in X\) and \(\gamma\in\Gamma\): (D1) \(d(x,y)=0\) if and only if \(x=y\). (D2) \(d(x,y)=d(y,x)\). (D3) If \(d(x,y)\leq\gamma\) and \(d(y,z)\leq\gamma\), then \(d(x,z)\leq\gamma\). The triple \((X,d,\Gamma)\) is called an ultrametric space. In this paper the author considers ultrametric spaces with a totally ordered value set \(\Gamma\). In this case, (D3) takes the simpler form (D3') \(d(x,z)\leq\max\{d(x,y),d(y,z)\}\). The author proves that a strictly contracting mapping of a spherically complete ultrametric space has a unique fixed point. Under certain assumptions, the author finds out that it is possible to cut the pseudo-convergent family in such a way that in an appropriate subspace the initial part of the family is a Cauchy family which has the fixed point as its limit, and then studies the application of the results to valued fields. Results by Kaplansky and a generalization of his crucial lemma for the uniqueness of maximal immediate extensions are used to prove that any subfield of a maximal valued field \(K\) has a spherical completion in \(K\). Finally, the results are applied to approximate roots of polynomials that satisfy Hensel's Lemma.
scientific article; zbMATH DE number 6680054
scientific article