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NumericalImplicitization :: ConvertToCone

ConvertToCone -- specifies whether to convert image to a cone

Synopsis

Description

This option specifies whether to replace the image $F(V(I))$ with the cone over $F(V(I))$. If true, then internally the target variety is treated as the affine cone over its projective closure - to be precise, the map $F$ is replaced with $t[F, 1]$, where $t$ is a new variable. The default value is false.

Since numericalHilbertFunction works by interpolating monomials (and thus only finds graded relations in the ideal of the image), this option is necessary when the map is not homogeneous. The following example demonstrates this for an affine rational curve.

i1 : R = CC[t]

o1 = R

o1 : PolynomialRing
 
i2 : F = {t, t^4, t^6}

          4   6
o2 = {t, t , t }

o2 : List
 
i3 : I = ideal 0_R

o3 = ideal 0

o3 : Ideal of R
 
i4 : (numericalHilbertFunction(F, I, 3, Verbose => false)).hilbertFunctionValue == 0
-- warning: experimental computation over inexact field begun
--          results not reliable (one warning given per session)

o4 = true
 
i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
Sampling image points ...
     -- used .0117299 seconds
Creating interpolation matrix ...
     -- used .00937079 seconds
Performing normalization preconditioning ...
     -- used .00341299 seconds
Computing numerical kernel ...
     -- used .000376549 seconds

o5 = a "numerical interpolation table", indicating
     the space of degree 3 forms in the ideal of the image has dimension 3

o5 : NumericalInterpolationTable
 
i6 : extractImageEquations(T, AttemptZZ => true)

o6 = | -y_0^2y_2+y_1^2y_3 -y_1^3+y_2^2y_3 -y_0^2y_1+y_2y_3^2 |

                          1                    3
o6 : Matrix (CC  [y ..y ])  <--- (CC  [y ..y ])
               53  0   3            53  0   3

See also

Functions with optional argument named ConvertToCone :

For the programmer

The object ConvertToCone is a symbol.