algcurves[singularities] - The singularities of an algebraic curve
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Calling Sequence
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singularities(f, x, y)
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Parameters
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f
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a polynomial specifying an algebraic curve
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x, y
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variables
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Description
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Let f be a squarefree polynomial in x and y. Then f defines an algebraic curve in the plane C^2, and also in the projective plane P^2 by making f homogeneous. This procedure computes the singular points of the curve in the projective plane. The points are given by homogeneous co-ordinates [X,Y,Z].
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The output of this procedure is a set consisting of lists of the following form .
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The genus of a curve is the number (d-1)*(d-2)/2 - Sum(delta invariants) where is the degree of the curve. Note that if we apply this formula to compute the genus, then for each singularity we must multiply the delta invariant by the degree of the algebraic extension over which the singularity is defined, because only one singularity of each conjugacy class is given in the output.
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Examples
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>
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Note that the conjugate (replace by is also a singularity. So the genus is (5-1)*(5-2)/2-1-1-1-2*1=1
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