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

Abstract

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 or 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.