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

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

2010-03-29

arxiv.org/abs/1003.5677v1

Mathematics

Commutative Algebra

47 pages

Scientific paper

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.

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