The Cab Curve --- a generalization of the Weierstrass form to arbitrary plane curves

(Japanese version)

When we construct one-point algebraic geometry codes, we have to find a basis of L(mQ) having pairwise distinct pole orders at Q. If we have the defining equation of a hyperelliptic curve in the Weierstrass form, then we can easily find such a basis of L(mQ) for the unique place Q at infinity. This fact can be generalized for an arbitrary curve as follows. The formal English reference for this fact is this research article.
Let V be an plane algebraic set defined by a bivariate polynomial of form \[ c_{b,0} X^b + c_{0,a} Y^a + \sum_{ai + bj < ab} c_{i,j} X^i Y^j. \] Then V is an algebraic curve with a unique rational place Q, and pole divisors of X and Y are aQ and bQ respectively. If V is nonsingular, then a basis of L(mQ) is \[ \{ X^i Y^j | 0 \leq i, 0 \leq j \leq a-1, ai + bj \leq m \}. \] Elements in the basis have pairwise distinct discrete valuations at Q.
More detailed statements and their complete proofs are described in AMS LaTeX 1.2 format, DVI format, PDF format, and PostScript format.

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