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.

NumericalInterpolationTable -- the class of all NumericalInterpolationTables

Description

See also

Functions and methods returning a numerical interpolation table :

Methods that use a numerical interpolation table :

For the programmer