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 .0130433 seconds Creating interpolation matrix ... -- used .0293417 seconds Performing normalization preconditioning ... -- used .0047085 seconds Computing numerical kernel ... -- used .00093799 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 |
The object ConvertToCone is a symbol.