Goppa Codes - Department of Mathematics

Goppa Codes Key One Chung
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Definition. Let F be the homogenization of f . Then the projective curve of F is
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�Cf := {(X0 : Y0 : Z0) ∈ P 2 (k)|F (X0, Y0, Z0) = 0}.
Example. f(x, y) = y − x 2 , g(x, y) = x − c. What is � Cf
F (X, Y, Z) = Z 2 (Y/Z − (x/Z) 2 ) = Y Z − X 2
G(X, Y, Z) = X − cZ
When Z = 1, Y = X 2 and X = c. So, we have (c : c 2 : 1).
When z = 0, X = 0 = X 2 . So, we have (0 : 1 : 0).
Therefore, | � Cf
� �Cg| = 2.
� �Cg?
Theorem (Bezout’s Theorem). If f, g ∈ k[x, y] are polynomials of degree d
and e respectively, then Cf and Cg intersect in at most de points. Further, � Cf
and � Cg intersect in exactly de points of P 2 (k), when points are counted with
multiplicity.
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