Finds (approximate) implicit degree d equations for the image of a variety. This is done via a numerical interpolation computation for the image.
If a numerical interpolation table has already been computed, then to avoid repetitive calculation one may run this function with the interpolation table as input.
We determine the defining quadrics of the twisted cubic. This is done with the option attemptExact, as follows:
i1 : R = CC[s,t]; |
i2 : F = {s^3,s^2*t,s*t^2,t^3}; |
i3 : extractImageEquations (F, ideal 0_R, 2, attemptExact => true) -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) Sampling image points ... -- used 0.012163 seconds Creating interpolation matrix ... -- used 0.0133457 seconds Performing normalization preconditioning ... -- used 0.135694 seconds Computing numerical kernel ... -- used 0.00105906 seconds o3 = {-2} | y_2^2-y_1y_3 | {-2} | y_1y_2-y_0y_3 | {-2} | y_1^2-y_0y_2 | 3 1 o3 : Matrix (CC [y , y , y , y ]) <--- (CC [y , y , y , y ]) 53 0 1 2 3 53 0 1 2 3 |
Here is how to do the same computation symbolically:
i4 : transpose gens ker map(QQ[s,t], QQ[y_0..y_3], {s^3,s^2*t,s*t^2,t^3}) o4 = {-2} | y_2^2-y_1y_3 | {-2} | y_1y_2-y_0y_3 | {-2} | y_1^2-y_0y_2 | 3 1 o4 : Matrix (QQ[y , y , y , y ]) <--- (QQ[y , y , y , y ]) 0 1 2 3 0 1 2 3 |
We determine the 5 Plücker quadrics defining the Grassmannian Gr(3,5).
i5 : R = CC[x_(1,1)..x_(3,5)]; I = ideal 0_R; o6 : Ideal of R |
i7 : F = (minors(3, genericMatrix(R, 3, 5)))_*; |
i8 : T = numericalHilbertFunction(F, I, 2, Verbose => false); |
i9 : extractImageEquations(T, attemptExact => true) o9 = {-2} | y_6y_7-y_5y_8+y_4y_9 | {-2} | y_3y_7-y_2y_8+y_1y_9 | {-2} | y_3y_5-y_2y_6+y_0y_9 | {-2} | y_3y_4-y_1y_6+y_0y_8 | {-2} | y_2y_4-y_1y_5+y_0y_7 | 5 1 o9 : Matrix (CC [y , y , y , y , y , y , y , y , y , y ]) <--- (CC [y , y , y , y , y , y , y , y , y , y ]) 53 0 1 2 3 4 5 6 7 8 9 53 0 1 2 3 4 5 6 7 8 9 |
The option attemptExact is experimental, and may result in inaccurate equations.