pairingMatrix -- Applies dual space functionals to polynomials

Synopsis

Usage:

M = pairingMatrix(S, D)
Inputs:
- S, an instance of the type PolySpace,
- D, an instance of the type DualSpace,
Outputs:
- M, a matrix, containing the values of the generators of D applied to the generators of S

Description

The dual space represents functionals from the polynomial ring to the base field. Given a polySpace S with n generators f_1,...,f_n and a dualSpace D with m generators p_1,...,p_m, pairingMatrix returns a nxm matrix M over the base field whose entries are p_j(f_i).

A dual functional is applied to a polynomial by taking the standard pairing of their coefficient vectors. In other words, the functional represented by the monomial a acts on monomials in the polynomial ring as a(a) = 1 and a(b) = 0 for all other monomials b.

i1 : R = CC[x,y];

i2 : S = polySpace matrix{{x+y,2*x+y^2}};

i3 : D = dualSpace(matrix{{1,x,y}}, origin R);

i4 : M = pairingMatrix(S, D)

o4 = {-1} | 0 1 1 |
     {-2} | 0 2 0 |

             2      3
o4 : Matrix R  <-- R

The function pairingMatrix can also be called with one or both inputs a ring element. If both arguments are single elements, the output is also a ring element rather than a matrix.

i5 : pairingMatrix(S, 1+x)

o5 = {-1} | 1 |
     {-2} | 2 |

             2      1
o5 : Matrix R  <-- R

i6 : pairingMatrix(x, D)

o6 = {-1} | 0 1 0 |

             1      3
o6 : Matrix R  <-- R

i7 : pairingMatrix(x, 1+x)

o7 = 1

o7 : R

Ways to use pairingMatrix :

pairingMatrix(PolySpace,DualSpace)
pairingMatrix(PolySpace,PolySpace)
pairingMatrix(PolySpace,RingElement)
pairingMatrix(RingElement,DualSpace)
pairingMatrix(RingElement,RingElement)

For the programmer

The object pairingMatrix is a method function.