# 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) -- warning: experimental computation over inexact field begun -- results not reliable (one warning given per session) 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

## 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.