Discrete Algebraic Methods

Proof: Let and for p, q, u, υ ∈ k[x, y]. Then, pυ coincides with uq at infinitely many points P ∈ E (k). By Lemma 5.2, we see that pυ = uq and thus r = t. If r = p/q for p, q ∈ k[x, y], then we can multiply both the numerator and the denominator by q. This shows that every rational function r(x, y) can be written as

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with u(x), υ(x), w(x) ∈ k[x] and w(x) ≠ 0.

The order of a point (see Theorem 5.3) is a concept which can also be applied to rational ...