The genus of an irreducible algebraic curve is a non-negative integer. It equals the dimension of the holomorphic differentials. It also equals (d-1)(d-2)/2 minus the sum of the delta invariants, which can be computed with algcurves[singularities]. Here d is the degree of the curve.

The polynomial f must be squarefree and have degree at least 1, otherwise an error message follows. A complete irreducibility check is not performed, only a few partial tests.

$\mathrm{with}\left(\mathrm{algcurves}\right)\:$

$f≔x^4+x^{2}y+y^2$

$f≔x^4+x^{2}y+y^2$

$\mathrm{factor}\left(f\right)$

$x^4+x^{2}y+y^2$

$\mathrm{genus}\left(f\,x\,y\right)$

Warning, negative genus so the curve is reducible

$\mathrm{-1}$

$\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$

$\left(x^2+\left(\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+3\right)}{2}+\frac{1}{2}\right)y\right)\left(x^2+\left(-\frac{\mathrm{RootOf}\left({\mathrm{\_Z}}^2+3\right)}{2}+\frac{1}{2}\right)y\right)$

$f≔\mathrm{subs}\left(z=1\,761328152x^6{z}^4-5431439286x^2{y}^8+2494x^2{z}^8+228715574724x^6{y}^4+9127158539954x^{10}-15052058268x^6{y}^2{z}^2+3212722859346x^8{y}^2-134266087241x^8{z}^2-202172841y^8{z}^2-34263110700x^4{y}^6-6697080y^6{z}^4-2042158x^4{z}^6-201803238y^{10}+12024807786x^4{y}^4{z}^2-128361096x^4{y}^2{z}^4+506101284x^2{z}^2{y}^6+47970216x^2{z}^4{y}^4+660492x^2{z}^6{y}^2-z^{10}-474z^8{y}^2-84366z^6{y}^4\right)\:$

$\mathrm{genus}\left(f\,x\,y\right)$

$10$