D.4.1.5 alg_kernel

Procedure from library algebra.lib (see algebra_lib).

Usage:

alg_kernel(phi,pr[,s,c]); phi map to basering, pr preimage ring, s string (name of kernel in pr), c integer.

Return:

a string, the kernel of phi as string.
If, moreover, a string s is given, the algorithm creates, in the preimage ring pr the kernel of phi with name s.
Three different algorithms are used depending on c = 1,2,3. If c is not given or c=0, a heuristically best method is chosen. (algorithm 1 uses the preimage command)

Note:

Since the kernel of phi lives in pr, it cannot be returned to the basering. If s is given, the user has access to it in pr via s. The basering may be a quotient ring.

Example:

LIB "algebra.lib";
ring r = 0,(a,b,c),ds;
ring s = 0,(x,y,z,u,v,w),dp;
ideal I = x-w,u2w+1,yz-v;
map phi = r,I;                // a map from r to s:
alg_kernel(phi,r);            // a,b,c ---> x-w,u2w+1,yz-v
→ 0
ring S = 0,(a,b,c),ds;
ring R = 0,(x,y,z),dp;
qring Q = std(x-y);
ideal i = x, y, x2-y3;
map phi = S,i;                 // a map to a quotient ring
alg_kernel(phi,S,"ker",3);     // uses algorithm 3
→ a-b,b^3-b^2+c
setring S;                     // you have access to kernel in preimage
ker;
→ ker[1]=a-b
→ ker[2]=c-b2+b3