Holomorphic differentials of an algebraic curve
differentials(f, x, y, opt)
irreducible polynomial in x and y
optional argument to change the form of the output
This command computes a basis of the holomorphic differentials of an irreducible algebraic curve f. Every holomorphic differential is of the form px,y/∂∂y⁢f⁢dx where p⁡x,y is a polynomial in x,y of degree ≤d−3 . Here d=degree⁡f,x,y is the degree of the curve.
If f is irreducible, then the dimension of the holomorphic differentials equals the genus of the curve; in other words, nops(differentials(f,x,y)) = genus(f,x,y).
If f has no singularities, then p⁡x,y can be any polynomial in x,y of degree ≤d−3 . So then the genus equals the number of monomials in x,y of degree ≤d−3 , which is 12⁢d−1⁡d−2.
For a singular curve, each singularity poses delta (the delta-invariant) independent linear conditions on the coefficients of p⁡x,y. So the genus equals 12⁢d−1⁡d−2 minus the sum of the delta-invariants. If δ=m⁡m−12 where m is the multiplicity of the singularity, then the linear conditions are equivalent with p⁡x,y vanishing with multiplicity m-1 at that singularity. If m⁡m−12<δ, then additional linear conditions exist, which are computed using integral_basis.
The output of this command will be a basis for all px,y/∂∂y⁢f⁢dx , or a basis for all p⁡x,y, in case a fourth argument skip_dx is given.
f ≔ y4+x3⁢y3+x4
f ≔ x3⁢y3+x4+y4
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