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 funtion
(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.)
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Use R ::= Q[a,b,c];
X := a+b-3c;
Use S ::= Q[x,y];
F := RMap(x^2,2,y^2); -- syntax for defining a map: the n-th
-- indeterminate in the domain will be mapped to
-- the n-th 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
-------------------------------
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