Macaulay2 » Documentation
Packages » NumericalImplicitization :: NumericalInterpolationTable
next | previous | forward | backward | up | index | toc

# NumericalInterpolationTable -- the class of all NumericalInterpolationTables

## Description

This is a type of hash table storing the output of a polynomial interpolation computation, with the following keys:

- \bf hilbertFunctionArgument: the argument, $d$, to the Hilbert function
- \bf hilbertFunctionValue: the value of the Hilbert function at $d$
- \bf imagePoints: a (vertical) list of sample points on the image
- \bf interpolationBasis: a matrix consisting of the degree $d$ monomials
- \bf interpolationSVD: the singular value decomposition of the interpolation matrix
- \bf interpolationMatrix: the matrix obtained by evaluating degree $d$ monomials at the sample points
- \bf map: the map $F$, of which the image is under consideration

    i1 : R = CC[x_(1,1)..x_(2,4)];

    i2 : F = (minors(2, genericMatrix(R, 2, 4)))_*;

    i3 : T = numericalHilbertFunction(F, ideal 0_R, 2, Verbose => false)

    o3 = a "numerical interpolation table", indicating
         the space of degree 2 forms in the ideal of the image has dimension 1

    o3 : NumericalInterpolationTable

    i4 : (T.hilbertFunctionArgument, T.hilbertFunctionValue)

    o4 = (2, 1)

    o4 : Sequence

## See also

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

## Functions and methods returning a numerical interpolation table:

- `numericalHilbertFunction(List,Ideal,List,ZZ)` -- see numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety
- `numericalHilbertFunction(List,Ideal,ZZ)` -- see numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety
- `numericalHilbertFunction(Matrix,Ideal,List,ZZ)` -- see numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety
- `numericalHilbertFunction(Matrix,Ideal,ZZ)` -- see numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety
- `numericalHilbertFunction(RingMap,Ideal,List,ZZ)` -- see numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety
- `numericalHilbertFunction(RingMap,Ideal,ZZ)` -- see numericalHilbertFunction -- computes the values of the Hilbert function for the image of a variety

## Methods that use a numerical interpolation table:

- `extractImageEquations(NumericalInterpolationTable)` -- see extractImageEquations -- finds implicit equations in a fixed degree for the image of a variety
- `net(NumericalInterpolationTable)`

## For the programmer

The object NumericalInterpolationTable is a type, with ancestor classes HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/NumericalImplicitization/doc.m2:485:0.