algcurves
differentials
find a basis of the holomorphic differentials of an algebraic curve
Calling Sequence
Parameters
Description
Examples
differentials(f, x, y, opt)
f
-
irreducible polynomial in x and y
x
variable
y
opt
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 where is a polynomial in x,y of degree . Here 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 can be any polynomial in x,y of degree . So then the genus equals the number of monomials in x,y of degree , which is .
For a singular curve, each singularity poses delta (the delta-invariant) independent linear conditions on the coefficients of . So the genus equals minus the sum of the delta-invariants. If where m is the multiplicity of the singularity, then the linear conditions are equivalent with vanishing with multiplicity m-1 at that singularity. If , then additional linear conditions exist, which are computed using integral_basis.
The output of this command will be a basis for all , or a basis for all , in case a fourth argument skip_dx is given.
See Also
AIrreduc
algcurves[genus]
algcurves[singularities]
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