Computing Riemann-Roch Spaces in Algebraic Function Fields and Related Topics

We develop a simple and efficient algorithm to compute Riemann-Roch spaces of divisors in general algebraic function fields which does not use the Brill-Noether method of adjoints nor any series expansions. The basic idea also leads to an elementary proof of the Riemann-Roch theorem. We describe the connection to the geometry of numbers of algebraic function fields and develop a notion and algorithm for divisor reduction. An important application is to compute in the divisor class group of an algebraic function field.