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NumericalImplicitization :: isOnImage

isOnImage -- tests whether a point lies on the image of a variety

Synopsis

Description

This method determines if a point in the ambient target space lies on the image of a variety. This is done via computing a pseudo-witness set for the image.

If a pseudo-witness set has already been computed, then to avoid repetitive calculation one may run this function with the pseudo-witness set as input.

The following example determines whether a point lies on the Grassmannian Gr(3,5) of P2’s in P4, under its Plücker embedding in P9.

i1 : R = CC[x_(1,1)..x_(3,5)]; I = ideal 0_R;

o2 : Ideal of R
i3 : F = (minors(3, genericMatrix(R, 3, 5)))_*;
i4 : W = pseudoWitnessSet(F, I, Repeats => 2, Verbose => false);
i5 : q = first numericalImageSample(F, I)

o5 = q

o5 : Point
i6 : isOnImage(W, q)

o6 = true
i7 : isOnImage(W, point random(CC^1, CC^#F))

o7 = false
i8 : isOnImage(W, point{{1_CC,0,0,0,0,0,0,0,0,0}})
     -- used 1.00243 seconds

o8 = true

See also

Ways to use isOnImage :