The Fekete-Szegö theorem with splitting conditions on the projective line of positive characteristic p

The logarithmic capacity is a measure of size for sets in C, which has arithmetic consequences. It initially arose in classical potential theory, but occurs in several contexts (including probability and number theory), and goes under several names, including Transfinite diameter, Chebychev constant, and exponential of the Robin constant. If the logarithmic capacity of a set can be computed, the Fekete-Szeg theorem gives a finiteness criterion/strong existence in arithmetic geometry [8]. Namely, whether the capacity is < 1 or > 1 determines the finiteness (or infiniteness) of the collection of algebraic integers whose conjugates lie near the set.In this dissertation, we are concerned with capacities for adelic sets in the positive characteristic projective line. Let K be a function field in one variable over a finite field of constants, with characteristic p > 0, and let K be its algebraic closure....