Polynomial factorization using Brill-Noether algorithm

Abstract

We give an algorithm to factorize a polynomial F of k[X,Y], where k is a perfect field. We define a projective plane curve C associated to F and we construct basis for k-vector spaces M(D), D beeing a divisor on the smooth projective model of C. This can be done using the generalization of Brill-Noether algorithm to the non-irreducible case. We recall that, in coding theory, this algorithm constructs the generator matrices (or the check matrices) of algebraic-geometric Goppa codes associated to absolutely irreducible projective plane curves.