map(Ring,Ring,Matrix) -- make a ring map

map(Ring,Ring,Matrix) -- make a ring map

Synopsis:

Usage: f = map(R,S,m)

Function: map -- make a map

Input:

R, an instance of class Ring: the target ring

S, an instance of class Ring: the source ring

m, an instance of class Matrix: a 1 by n matrix over R, where n is the number of variables in the polynomial ring S, or a matrix over the common coefficient ring of the two rings.

Output:

f, an instance of class RingMap: the ring homomorphism from S to R which, in case m is a matrix over R, sends the i-th variable of S to the i-th entry in m, or, in case m is a matrix over the common coefficient ring, is the linear change of coordinates corresponding to m.

Optional arguments :

map(..., Degree)

map(..., DegreeMap => ...)

i1 : R = ZZ[x,y];

i2 : S = ZZ[a,b,c];

i3 : f = map(R,S,matrix {{x^2,x*y,y^2}})



               2        2

o3 = map(R,S,{x , x*y, y })



o3 : RingMap R <--- S

i4 : f(a+b+c^2)



      4    2

o4 = y  + x  + x*y



o4 : R

i5 : g = map(R,S,matrix {{1,2,3},{4,5,6}})



o5 = map(R,S,{x + 4y, 2x + 5y, 3x + 6y})



o5 : RingMap R <--- S

i6 : g(a+b)



o6 = 3x + 9y



o6 : R

If the coefficient ring of S is itself a polynomial ring, then one may optionally include values to which its variables should be sent: they should appear first in the matrix m.

i7 : S = ZZ[a][b,c];

i8 : h = map(S,S,matrix {{b,c,a}})



o8 = map(S,S,{b, c, a})



o8 : RingMap S <--- S

i9 : h(a^7 + b^3 + c)



      7    3

o9 = b  + c  + a



o9 : S

i10 : k = map(S,S,matrix {{c,b}})



o10 = map(S,S,{a, c, b})



o10 : RingMap S <--- S

i11 : k(a^7 + b^3 + c)



       3        7

o11 = c  + b + a



o11 : S