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.