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

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.