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

## Synopsis

• Usage:
(A,stds) = affinePointsMat(M,R)
• Inputs:
• M, , in which each column consists of the coordinates of a point
• R, , coordinate ring of the affine space containing the points
• Outputs:
• A, , standard monomials evaluated on points
• stds, , 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.