Berkovich space

In mathematics, a Berkovich space, introduced by Berkovich (1990), is an analogue of an analytic space for p-adic geometry, refining Tate's notion of a rigid analytic space.

Berkovich spectrum

A seminorm on a ring A is a non-constant function f→|f| from A to the non-negative reals such that |0| = 0, |1| = 1, |f + g| ≤ |f| + |g|, |fg| ≤ |f||g|. It is called multiplicative if |fg| = |f||g| and is called a norm if |f| = 0 implies f = 0.

If A is a normed ring with norm f  ||f|| then the Berkovich spectrum of A is the set of multiplicative seminorms || on A that are bounded by the norm of A. The Berkovich spectrum is topologized with the weakest topology such that for any f in A the map taking || to |f| is continuous..

The Berkovich spectrum of a normed ring A is non-empty if A is non-zero and is compact if A is complete.

The spectral radius ρ(f) = lim |fn|1/n of f is equal to supx|f|x

Examples

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