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algcurves

 differentials
 Holomorphic differentials of an algebraic curve

 Calling Sequence differentials(f, x, y, opt)

Parameters

 f - irreducible polynomial in x and y x - variable y - variable opt - optional argument to change the form of the output

Description

 • 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 $p\left(x,y\right)$. 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 $p\left(x,y\right)$ 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 $p\left(x,y\right)$, in case a fourth argument skip_dx is given.

Examples

 > with(algcurves):
 > f:=y^4+x^3*y^3+x^4;
 ${f}{≔}{{x}}^{{3}}{}{{y}}^{{3}}{+}{{x}}^{{4}}{+}{{y}}^{{4}}$ (1)
 > differentials(f,x,y);
 $\left[\frac{{x}{}{\mathrm{dx}}}{{3}{}{{x}}^{{3}}{+}{4}{}{y}}{,}\frac{{{x}}^{{2}}{}{\mathrm{dx}}}{{y}{}\left({3}{}{{x}}^{{3}}{+}{4}{}{y}\right)}\right]$ (2)
 > differentials(f,x,y,skip_dx);
 $\left[{x}{}{{y}}^{{2}}{,}{{x}}^{{2}}{}{y}\right]$ (3)
 > nops((3));
 ${2}$ (4)
 > genus(f,x,y);
 ${2}$ (5)