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NumericalImplicitization :: numericalImageDim

numericalImageDim -- computes the dimension of the image of a variety

Synopsis

Description

The method computes the dimension of the image of a variety numerically. Even if the source variety and map are projective, the affine (Krull) dimension is returned. This ensures consistency with dim.

The following example computes the affine dimension of 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)];
i2 : F = (minors(3, genericMatrix(R, 3, 5)))_*;
i3 : numericalImageDim(F, ideal 0_R)

o3 = 7

For comparison, here is how to do the same computation symbolically.

i4 : R = QQ[x_(1,1)..x_(3,5)];
i5 : F = (minors(3, genericMatrix(R, 3, 5)))_*;
i6 : dim ker map(R,QQ[y_0..y_(#F-1)],F)

o6 = 7

Next is an example where direct symbolic computation fails to terminate quickly. Part of the Alexander-Hirschowitz theorem states that the 14th secant variety of the 4th Veronese of P4 has affine dimension 69, rather than the expected 14*4 + 13 + 1 = 70. See J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables, J. Alg. Geom. 4(2) (1995), 201-222. We numerically verify this below.

i7 : R = CC[a_(1,1)..a_(14,5)];
i8 : F = sum(1..14, i -> basis(4, R, Variables=>toList(a_(i,1)..a_(i,5))));
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)

             1       70
o8 : Matrix R  <--- R
i9 : time numericalImageDim(F, ideal 0_R)
     -- used 0.0810486 seconds

o9 = 69

Ways to use numericalImageDim :