Finding a Basis of a Linear System with Pairwise Distinct Discrete Valuations on an Algebraic Curve

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 convenient 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 Gröbner basis for the ideal defining the curve.