By Baker M., Rumely R.
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Extra resources for Potential theory on the Berkovich projective line
1. THE BERKOVICH AFFINE LINE A1Berk 21 on K r−1 T to the continuous multiplicative seminorm [ ]ιr,R (x) on K R−1 T defined by [f ]ιr,R (x) = [π(f )]x , where π : K R−1 T → K r−1 T is the natural K-algebra homomorphism. This map is easily checked to be continuous, with dense image, so ιr,R is injective. It thus makes sense to consider the direct limit, or union, of the discs D(0, R). 1) D(0, R) . 1) by specifying continuous maps in each direction which are inverse to one another. On the one hand, for each R there is an obvious map ιR : D(0, R) → A1Berk coming from the inclusion K[T ] → K R−1 T , and if r < R then ιr = ιR ◦ ιr,R .
C) [f + g]x ≤ max([f ]x , [g]x ), with equality if [f ]x = [g]x . Proof. (A) For each n, ([f ]x )n = [f n ]x ≤ Cx f n = Cx f n , so [f ]x ≤ f , and letting n → ∞ gives the desired inequality. (B) By the definition of the Gauss norm, c = |c|. If c = 0 then trivially [c]x = 0; otherwise, [c]x ≤ c = |c| and [c−1 ]x ≤ c−1 = |c−1 |, 1/n Cx 1 2 1. THE BERKOVICH UNIT DISC while multiplicativity gives [c]x · [c−1 ]x = [c · c−1 ]x = 1. Combining these gives [c]x = |c|. (C) The binomial theorem shows that for each n, n ([f + g]x )n = [(f + g)n ]x = [ k=0 n ≤ | k=0 n k n−k f g ]x k n | · [f ]kx [g]xn−k ≤ k n [f ]kx [g]xn−k k=0 ≤ (n + 1) · max([f ]x , [g]x )n .
Hence [T − a]D(ai ,ri ) := |z − a| = |a − ai | = [T − a]x . 4) holds in this case as well. 5) [f ]x = lim [f ]D(ai ,ri ) . i→∞ Now suppose the family F has non-empty intersection, and let a be a point in that intersection. 4) gives [T − a]x = lim [T − a]D(ai ,ri ) ≤ lim ri = r , i→∞ i→∞ while the definition of r shows that [T − a]x ≥ r. Thus [T − a]x = r. Hence the disc D(a, r) (which may consist of a single point if r = 0) is a minimal element of F. 5) holds for any sequence of discs D(ai , ri ) such that ri = [T − ai ]x satisfies lim ri = r.