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algcurves

  

differentials

  

Holomorphic differentials of an algebraic curve

 

Calling Sequence

Parameters

Description

Examples

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 px,y/yfdx where px,y is a polynomial in x,y of degree d3 . Here d=degreef,x,y 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 px,y can be any polynomial in x,y of degree d3 . So then the genus equals the number of monomials in x,y of degree d3 , which is 12d1d2.

• 

For a singular curve, each singularity poses delta (the delta-invariant) independent linear conditions on the coefficients of px&comma;y. So the genus equals 12d1d2 minus the sum of the delta-invariants. If δ=mm12 where m is the multiplicity of the singularity, then the linear conditions are equivalent with px&comma;y vanishing with multiplicity m-1 at that singularity. If mm12<δ, then additional linear conditions exist, which are computed using integral_basis.

• 

The output of this command will be a basis for all px,y/yfdx , or a basis for all px&comma;y, in case a fourth argument skip_dx is given.

Examples

withalgcurves&colon;

fy4&plus;x3y3&plus;x4

f:=x3y3&plus;x4&plus;y4

(1)

differentialsf&comma;x&comma;y

xdx3x3&plus;4y&comma;x2dxy3x3&plus;4y

(2)

differentialsf&comma;x&comma;y&comma;skip_dx

xy2&comma;x2y

(3)

nops

2

(4)

genusf&comma;x&comma;y

2

(5)

See Also

AIrreduc

algcurves[genus]

algcurves[singularities]

 


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