If one defines analytical functions on the p-adic straight line or, more generally, on -manifolds, as those which can be represented locally by convergent power series , then all functions are analytic, because they are totally disconnected . In order to be able to define a meaningful concept of analytical functions, points are added that make the space coherent .
For is , because all multiplicative semi-norms are given by for a .
For an algebraically closed, complete, non-Archimedean field, multiplicative semi-norms are either of the form
for one or
for a . Here referred to .
Berkovich's Classification Theorem
Each corresponds to a descending sequence of nested enclosed spheres . With you get the following classification into four types:
Type I: for a
Type II: for one
Type III: for one
Type IV:
properties
The Berkovich straight line is a locally compact Hausdorff space . The type I points corresponding to the points in are close to . The Berkovich line is clearly path-connected; That is, every two points can be connected by an unambiguous shortest path. Type II points are branch points.
literature
V. Berkovich: Spectral theory and analytic geometry over non-Archimedean fields , American Mathematical Society, Mathematical Surveys and Monographs 33, 1990