Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Maps on ultrametric spaces, Hensel's Lemma, and differential equations
  over valued fields   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.

4
The categories of this paper are: cond-mat
1
The categories of this paper are: cs
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg
0
The categories of this paper are: hep-ph
9
The categories of this paper are: nlin
12
The categories of this paper are: nucl-th
28
The categories of this paper are: alg-geom
10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
11
The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
21
The categories of this paper are: adap-org
24
The categories of this paper are: dg-ga
6
The categories of this paper are: astro-ph
18
The categories of this paper are: econ