affinePointsMat -- produces the matrix of values of the standard monomials on a set of points

Synopsis

Usage:

(A,stds) = affinePointsMat(M,R)
Inputs:
- M, a matrix, in which each column consists of the coordinates of a point
- R, a polynomial ring, coordinate ring of the affine space containing the points
Outputs:
- A, a matrix, standard monomials evaluated on points
- stds, a matrix, whose entries are the standard monomials

Description

This function uses the Buchberger-Moeller algorithm to compute a the matrix A in which the columns are indexed by standard monomials, the rows are indexed by points, and the entries are given by evaluation. The ordering of the standard monomials is recorded in the matrix stds which has a single column. Here is a simple example.

i1 : M = random(ZZ^3, ZZ^5)

o1 = | 8 7 3 8 8 |
     | 1 8 7 5 5 |
     | 3 3 8 7 2 |

              3        5
o1 : Matrix ZZ  <--- ZZ

i2 : R = QQ[x,y,z]

o2 = R

o2 : PolynomialRing

i3 : (A,stds) = affinePointsMat(M,R)

o3 = (| 1 3 1 8 9  |, {0}  | 1  |)
      | 1 3 8 7 9  |  {-1} | z  |
      | 1 8 7 3 64 |  {-1} | y  |
      | 1 7 5 8 49 |  {-1} | x  |
      | 1 2 5 8 4  |  {-2} | z2 |

o3 : Sequence

Caveat

Program does not check that the points are distinct.

Ways to use `affinePointsMat` :

"affinePointsMat(Matrix,Ring)"

For the programmer

The object affinePointsMat is a method function.