Computing the genus
Write again f (x, y) =
P
i,j ai,j xj
yi
.
Theorem (Novocin)
The ωi,j = xj−1yi−1
∂f
∂y
dx, i, j ∈ N, are holomorphic at the finite
nonsingular points. Any holomorphic differential on C is a
linear combination of the ωi,j for (i, j) strictly in the convex
hull of the support of f (x, y).
⇝ Strategy: Compute local parametrisations at all the
singular points and at the points at infinity. Plug them into
the ωi,j , and use linear algebra over K to find the
combinations whose polar parts vanish.
We get a K-basis of the space of holomorphic differentials.
The genus of the curve is its dimension.
Nicolas Mascot Algebraic curves