Integral closure (Preparation for Riemann-Roch)
Let K(C) = Frac K(x)[y]/f (x, y).
The integral closure of K[x] in K(C) is
OC = {h(x, y) ∈ K(C) | h holomorphic above x ̸= ∞}.
Start with the approximation O =
L
j<n K[x]yj
1,
where y1 = lcy (f )y.
For all irreducible U(x) ∈ K[x], O is U-maximal unless
U2
| discy f (x, y).
For such U, compute the parametrisations at the points above
U(x) = 0, plug them into the xi
yj
1/U(x) for i < deg U and
j < n, and find linear combinations whose polar parts vanish.
Then join the local bases by performing a HNF over K[x].
Nicolas Mascot Algebraic curves