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4.8.12 Ring Mappings: the Image Function

The function
Image
implements a ring homomorphism. Suppose S is
the current ring and R is another ring. If X is an object in R, the
function
Image
may be used to substitute polynomials in S for the
indeterminates in X. An example is given below and complete details
are given in the online help entry for
Image
.
To make substitutions within a single ring, one would usually use
Eval
or
Subst
rather than
Image
. To map a polynomial or ideal
from an outside ring into the current ring, the functions
QZP and
ZPQ are sometimes useful. To map a polynomial or rational function
(or a list, matrix, or vector of these) from R to S without changing
indeterminates, use the function
BringIn
. (
BringIn is only
applicable if the indeterminates of the object to be mapped are a
subset of those in S.)
Use R ::= QQ[a,b,c];
X := a+b3c;
Use S ::= QQ[x,y];
F := RMap(x^2,2,y^2);  syntax for defining a map: the nth
 indeterminate in the domain will be mapped to
 the nth element listed in RMap.
X;  X lives in the ring R
R :: a + b  3c

Image(X, F);  the image of E under the map F
x^2  3y^2 + 2

Image(R:: (a+b)^2, F);
x^4 + 4x^2 + 4

