Singular

7.10.3.7 lpSubstitute

Usage:

lpSubstitute(f,s1,s2[,G]); f poly, s1 list (ideal) of variables to replace, s2 list (ideal) of polynomials to replace with, G optional ideal to reduce with.

Return:

poly, the substituted polynomial

Assume:

- basering is a Letterplace ring
- s1 contains a subset of the set of variables
- s2 and s1 are of the same size
- G is a Groebner basis,
- the current ring has a sufficient degbound (which also can be calculated with lpCalcSubstDegBound())

Note:

the procedure implements the image of a polynomial f
under an endomorphism of a free algebra, defined by s1 and s2: variables, not present in s1, are left unchanged;
variable s1[k] is mapped to a polynomial s2[k].
- An optional ideal G extends the endomorphism as above to the morphism into the factor algebra K<X>/G.

LIB "fpaprops.lib";
ring r = 0,(x,y,z),dp;
def R = freeAlgebra(r, 4);
setring R;
ideal G = x*y; // optional
poly f = 3*x*x+y*x;
ideal s1 = x, y;
ideal s2 = y*z*z, x; // i.e. x --> yzz and y --> x
// the substitution probably needs a higher degbound
int minDegBound = lpCalcSubstDegBound(f,s1,s2);
minDegBound; // thus the bound needs to be increased
==> 9
setring r; // back to original r
def R1 = freeAlgebra(r, minDegBound);
setring R1;
lpSubstitute(imap(R,f), imap(R,s1), imap(R,s2));
==> 3*y*z*z*y*z*z+x*y*z*z
// the last parameter is optional; above it was G=<xy>
// the output will be reduced with respect to G
lpSubstitute(imap(R,f), imap(R,s1), imap(R,s2), imap(R,G));
==> 3*y*z*z*y*z*z