APPROXIMATING GREEN'S FUNCTIONS ON ℙ1 POSITIVE CHARACTERISTIC

Abstract

Fix a finite K-symmetric set [Formula: see text] and a K-symmetric probability vector [Formula: see text]. Let 𝔇v be a finite union of balls [Formula: see text] for some ah ∈ Kv and some [Formula: see text], where the balls 𝔅(ah, rh) are disjoint from 𝔛. Put 𝔈v := 𝔇v ∩ ℙ1(Kv). Then there exists a positive integer Nv such that for each sufficiently large integer N divisible by Nv, there are a number Rv, with [Formula: see text], and an [Formula: see text]-function fv(z) ∈ Kv(z) of degree N whose zeros form a "well-distributed" sequence in 𝔈v such that [Formula: see text] is a disjoint union of balls centered at the zeros of fv(z) and for all z ∉ 𝔇v, [Formula: see text]