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 |