Maps on ultrametric spaces, Hensel's Lemma, and differential equations over valued fields

TL;DR

This paper develops criteria for maps on ultrametric spaces to be surjective and preserve spherical completeness, deriving Hensel's Lemma and the Implicit Function Theorem in valued fields, with applications to differential equations and power series fields.

Contribution

It introduces a unified criterion for surjectivity and spherical completeness preservation, leading to new proofs and extensions of Hensel's Lemma and the Implicit Function Theorem in valued fields.

Findings

Hensel's Lemma follows from the criterion in henselian fields

An infinite-dimensional Implicit Function Theorem is established

Criteria for solutions of differential equations in valued differential fields

Abstract

We give a criterion for maps on ultrametric spaces to be surjective and to preserve spherical completeness. We show how Hensel's Lemma and the multi-dimensional Hensel's Lemma follow from our result. We give an easy proof that the latter holds in every henselian field. We also prove a basic infinite-dimensional Implicit Function Theorem. Further, we apply the criterion to deduce various versions of Hensel's Lemma for polynomials in several additive operators, and to give a criterion for the existence of integration and solutions of certain differential equations on spherically complete valued differential fields, for both valued D-fields in the sense of Scanlon, and differentially valued fields in the sense of Rosenlicht. We modify the approach so that it also covers logarithmic-exponential power series fields. Finally, we give a criterion for a sum of spherically complete subgroups of a valued abelian group to be spherically complete. This in turn can be used to determine elementary properties of power series fields in positive characteristic.