genus - Maple Help

algcurves

 genus
 The genus of an algebraic curve

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

Parameters

 f - squarefree polynomial specifying an algebraic curve x, y - variables opt - (optional) a sequence of options

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{algcurves}\right):$
 > $f≔{x}^{4}+{x}^{2}y+{y}^{2}$
 ${f}{≔}{{x}}^{{4}}{+}{{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (1)
 > $\mathrm{factor}\left(f\right)$
 ${{x}}^{{4}}{+}{{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (2)
 > $\mathrm{genus}\left(f,x,y\right)$
 ${-1}$ (3)
 > $\mathrm{evala}\left(\mathrm{AFactor}\left(f\right)\right)$
 $\left({{x}}^{{2}}{+}\left(\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{3}\right)}{{2}}{+}\frac{{1}}{{2}}\right){}{y}\right){}\left({{x}}^{{2}}{+}\left({-}\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{3}\right)}{{2}}{+}\frac{{1}}{{2}}\right){}{y}\right)$ (4)
 > $f≔\mathrm{subs}\left(z=1,761328152{x}^{6}{z}^{4}-5431439286{x}^{2}{y}^{8}+2494{x}^{2}{z}^{8}+228715574724{x}^{6}{y}^{4}+9127158539954{x}^{10}-15052058268{x}^{6}{y}^{2}{z}^{2}+3212722859346{x}^{8}{y}^{2}-134266087241{x}^{8}{z}^{2}-202172841{y}^{8}{z}^{2}-34263110700{x}^{4}{y}^{6}-6697080{y}^{6}{z}^{4}-2042158{x}^{4}{z}^{6}-201803238{y}^{10}+12024807786{x}^{4}{y}^{4}{z}^{2}-128361096{x}^{4}{y}^{2}{z}^{4}+506101284{x}^{2}{z}^{2}{y}^{6}+47970216{x}^{2}{z}^{4}{y}^{4}+660492{x}^{2}{z}^{6}{y}^{2}-{z}^{10}-474{z}^{8}{y}^{2}-84366{z}^{6}{y}^{4}\right):$

This f is a polynomial of degree 10 having a maximal number of cusps according to the Plucker formulas. It was found by Rob Koelman. It has 26 cusps and no other singularities, hence the genus is (10-1)*(10-2)/2 - 26 = 10.

 > $\mathrm{genus}\left(f,x,y\right)$
 ${10}$ (5)