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algcurves

  

genus

  

The genus of an algebraic curve

 

Calling Sequence

Parameters

Description

Examples

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;

fx4+x2y+y2

(1)

factor(f);

x4+x2y+y2

(2)

genus(f,x,y);

−1

(3)

evala(AFactor(f));

x2+RootOf_Z2+32+12yx2+RootOf_Z2+32+12y

(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)

See Also

AIrreduc

algcurves[differentials]

algcurves[parametrization]

algcurves[singularities]