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 \( \mathcal{L}(D) \) of a divisor D from an ideal basis of the ideal \( \mathcal{L}(D + \infty Q) \) of the affine coordinate ring \( \mathcal{L}(\infty Q) \) of the given algebraic curve, where \( \mathcal{L}(D + \infty Q): = \bigcup\nolimits_{i = 1}^\infty {\mathcal{L}(D + iQ)} \). 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 Gröbner basis of the ideal defining the curve.
2000 Mathematical Subject Classification. Primary 14Q05, 13P10; Secondary 94B27, 11T71, 14H05, 14C20.
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Matsumoto, R., Miura, S. (1999). Computing a Basis of \( \mathcal{L}(D) \) on an affine algebraic curve with one rational place at infinit. In: Fossorier, M., Imai, H., Lin, S., Poli, A. (eds) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes. AAECC 1999. Lecture Notes in Computer Science, vol 1719. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46796-3_27
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