Computing a Basis of $$ \mathcal{L}(D) $$ on an affine algebraic curve with one rational place at infinit

Abstract

Under the assumption that we have defining equations of an affine algebraic curve in special position with respect to a rational place Q, we propose an algorithm computing a basis of L(D) of a divisor D from an ideal basis of the ideal L(D + ∞Q) of the affine coordinate ring L(∞Q) of the given algebraic curve, where L(D + ∞Q) := ∪i=1∞L(D + iQ). Elements in the basis produced by our algorithm have pairwise distinct discrete valuations at Q, which is crucial in the construction of algebraic geometry codes. Our method is applicable to a curve embedded in an affine space of arbitrary dimension, and involves only the Gaussian elimination and the division of polynomials by the Grobner basis of the ideal defining the curve.