affineMakeRingMaps -- evaluation on points

Synopsis

Usage:

affineMakeRingMaps(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 list, of ring maps corresponding to evaluations at each point

Description

Giving the coordinates of a point in affine space is equivalent to giving a ring map from the polynomial ring to the ground field: evaluation at the point. Given a finite collection of points encoded as the columns of a matrix, this function returns a corresponding list of ring maps.

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 : phi = affineMakeRingMaps(M,R)

o3 = {map (QQ, R, {8, 1, 3}), map (QQ, R, {7, 8, 3}), map (QQ, R, {3, 7, 8}),
     ------------------------------------------------------------------------
     map (QQ, R, {8, 5, 7}), map (QQ, R, {8, 5, 2})}

o3 : List

i4 : apply (gens(R),r->phi#2 r)

o4 = {3, 7, 8}

o4 : List

i5 : phi#2

o5 = map (QQ, R, {3, 7, 8})

o5 : RingMap QQ <-- R

Ways to use affineMakeRingMaps :

affineMakeRingMaps(Matrix,Ring)

For the programmer

The object affineMakeRingMaps is a method function.