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NumericalImplicitization :: numericalImageDim

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

Synopsis

Description

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 computes the affine dimension of the Grassmannian Gr(3,5) of P2’s in P4, under its Plücker embedding.

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

Here 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 4 has affine dimension 69, rather than the expected 14*4 + 13 + 1 = 70. 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 0.107478 seconds

o9 = 69

Reference

[1] J. Alexander, A. Hirschowitz, Polynomial interpolation in several variables. J. Alg. Geom. 4 (2) (1995), 201-222.

Ways to use numericalImageDim :