Ring Mappings: the Image Function

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.)

Example

  Use R ::= QQ[a,b,c];
  X := a+b-3c;
  Use S ::= QQ[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
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  Image(X, F); -- the image of E under the map F
x^2 - 3y^2 + 2
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  Image(R:: (a+b)^2, F);
x^4 + 4x^2 + 4
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