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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

 > with(algcurves):
 > f:=x^4+x^2*y+y^2;
 ${f}{≔}{{x}}^{{4}}{+}{{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (1)
 > factor(f);
 ${{x}}^{{4}}{+}{{x}}^{{2}}{}{y}{+}{{y}}^{{2}}$ (2)
 > genus(f,x,y);
 ${-1}$ (3)
 > evala(AFactor(f));
 $\left({{x}}^{{2}}{+}\left({-}\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{3}\right)}{{2}}{+}\frac{{1}}{{2}}\right){}{y}\right){}\left({{x}}^{{2}}{+}\left(\frac{{\mathrm{RootOf}}{}\left({{\mathrm{_Z}}}^{{2}}{+}{3}\right)}{{2}}{+}\frac{{1}}{{2}}\right){}{y}\right)$ (4)
 > f:=subs(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):

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.

 > genus(f,x,y);
 ${10}$ (5)