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# ConvertToCone -- specifies whether to convert image to a cone

- Usage:
    `numericalHilbertFunction(..., ConvertToCone => false)`

## 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

    o4 = true

    i5 : T = numericalHilbertFunction(F, I, 3, ConvertToCone => true)
    Sampling image points ...
         -- used .107342 seconds
    Creating interpolation matrix ...
         -- used .00558085 seconds
    Performing normalization preconditioning ...
         -- used .00268619 seconds
    Computing numerical kernel ...
         -- used .00037124 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

- numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety

## Functions with optional argument named ConvertToCone:

- `numericalHilbertFunction(...,ConvertToCone=>...)`

## For the programmer

The object ConvertToCone is a symbol.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NumericalImplicitization/doc.m2:956:0.