Finding a Basis of a Linear System with Pairwise Distinct Discrete Valuations on an Algebraic Curve
✍ Scribed by Ryutaroh Matsumoto; Shinji Miura
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 358 KB
- Volume
- 30
- Category
- Article
- ISSN
- 0747-7171
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✦ Synopsis
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.