Rational functions with given ramification in characteristic p

Abstract

Using limit linear series and a result controlling degeneration from separable maps to inseparable maps, we give a formula for the number of rational functions (up to automorphism of the target) on the projective line with ramification to order e_i at general points P_i, in the case that all e_i are less than the characteristic. Unlike the case of characteristic 0, the answer is not given by Schubert calculus, nor is the number of maps always finite for distinct P_i, even in the tamely ramified case. However, finiteness for general P_i, obtained by exploiting the relationship to branched covers, is a key part of the argument.