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NumericalImplicitization :: numericalImageSample

numericalImageSample -- samples general points on the image of a variety

Synopsis

Description

Computes a list of sample points on the image of a variety numerically. This function calls numericalSourceSample.

If s is unspecified, then it is assumed that s = 1. In this case, the single point is returned, rather than a list.

The following samples a point from the twisted cubic. We then independently verify that this point does lie on the twisted cubic.

i1 : R = CC[s,t];
i2 : F = {s^3,s^2*t,s*t^2,t^3};
i3 : p = numericalImageSample(F, ideal 0_R)

o3 = p

o3 : Point
i4 : A = matrix{p#Coordinates_{0,1,2}, p#Coordinates_{1,2,3}};

                2          3
o4 : Matrix CC    <--- CC
              53         53
i5 : numericalRank A == 1

o5 = true

Here is how to sample a point from the Grassmannian Gr(3,5) of P2’s in P4, under its Plücker embedding. We take maximal minors of a 3 x 5 matrix, whose row span gives a P2 in P4.

i6 : R = CC[x_(1,1)..x_(3,5)];
i7 : F = (minors(3, genericMatrix(R, 3, 5)))_*;
i8 : numericalImageSample(F, ideal 0_R)

o8 = {.235327-.159959*ii, -.583293+.111338*ii, .0110037+.269476*ii, .535607-.134897*ii, -.688164+.914162*ii, -.0166354+.170936*ii, .461736-.724404*ii, .565043+.697799*ii, .378877-.203586*ii, .496259+.433603*ii}

o8 : Point

See also

Ways to use numericalImageSample :