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makeRingMaps -- evaluation on points

Synopsis

Usage:
makeRingMaps(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 = | 4 1 6 5 5 |
     | 3 3 5 5 7 |
     | 8 9 6 4 0 |

              3        5
o1 : Matrix ZZ  <--- ZZ

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

o2 = R

o2 : PolynomialRing

i3 : phi = makeRingMaps(M,R)

o3 = {map(QQ,R,{4, 3, 8}), map(QQ,R,{1, 3, 9}), map(QQ,R,{6, 5, 6}),
     ------------------------------------------------------------------------
     map(QQ,R,{5, 5, 4}), map(QQ,R,{5, 7, 0})}

o3 : List

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

o4 = {6, 5, 6}

o4 : List

i5 : phi#2

o5 = map(QQ,R,{6, 5, 6})

o5 : RingMap QQ <--- R

Ways to use `makeRingMaps` :

makeRingMaps(Matrix,Ring)