This command computes a basis of the holomorphic differentials of an irreducible algebraic curve f. Every holomorphic differential is of the form $\left(p\left(x,y\right)/\frac{\partial }{\partial y}f\right)\mathrm{dx}$ where $p\left(x\,y\right)$ is a polynomial in x,y of degree $\le d-3$ . Here $d=\mathrm{degree}\left(f\,\left\{x\,y\right\}\right)$ 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 $p\left(x\,y\right)$ can be any polynomial in x,y of degree $\le d-3$ . So then the genus equals the number of monomials in x,y of degree $\le d-3$ , which is $\frac{\left(d-1\right)\left(d-2\right)}{2}$.

For a singular curve, each singularity poses delta (the delta-invariant) independent linear conditions on the coefficients of $p\left(x\&comma;y\right)$. So the genus equals $\frac{\left(d-1\right)\left(d-2\right)}{2}$ minus the sum of the delta-invariants. If $\mathrm{\delta }=\frac{m\left(m-1\right)}{2}$ where m is the multiplicity of the singularity, then the linear conditions are equivalent with $p\left(x\&comma;y\right)$ vanishing with multiplicity m-1 at that singularity. If $\frac{m\left(m-1\right)}{2}<\mathrm{\delta }$, then additional linear conditions exist, which are computed using integral_basis.

The output of this command will be a basis for all $\left(p\left(x,y\right)/\frac{\partial }{\partial y}f\right)\mathrm{dx}$ , or a basis for all $p\left(x\&comma;y\right)$, in case a fourth argument skip_dx is given.

$\mathrm{with}\left(\mathrm{algcurves}\right)\&colon;$

$f≔y^4\&plus;x^3{y}^3\&plus;x^4$

$f≔x^3{y}^3+x^4+y^4$

$\mathrm{differentials}\left(f\&comma;x\&comma;y\right)$

$\left[\frac{x\mathrm{dx}}{3x^3+4y}\&comma;\frac{x^2\mathrm{dx}}{y\left(3x^3+4y\right)}\right]$

$\mathrm{differentials}\left(f\&comma;x\&comma;y\&comma;\mathrm{skip\_dx}\right)$

$\left[x{y}^2\&comma;x^{2}y\right]$

$\mathrm{nops}\left(\right)$

$2$

$\mathrm{genus}\left(f\&comma;x\&comma;y\right)$

$2$