ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

3. Number Theory and Arithmetic Algebraic Geometry

1991 MS Classification: 11R58, 14H05, 14Q05 Heß, Florian, TU Berlin, Germany Effective construction of Riemann-Roch spaces in global function fields Let F/k be a global function field, that is, a field F of transcendence degree one over a finite field k. The set of all places of F is denoted by ℙ_{F}. Given a divisor D = ∑_{𝔭∈ℙ_{F}} n_{𝔭}𝔭 the Riemann-Roch space of D is defined to be

L(D) = {α ∈ F | ν_{𝔭}(α) ≥ −n_{𝔭} for all 𝔭 ∈ ℙ_{F} }.

It is a finite dimensional vector space over k. Let now F/k explicitly be given by an equation f(x, y) = 0 with an irreducible, in y separable and monic polynomial f ∈ k[x, y] (which is always possible). In this contribution an effective algorithm for the construction of a basis of L(D) using the above representation of F is presented. There are many interesting applications, for example the construction of algebraic geometric codes.