Singular

D.4.2.6 is_injective

Usage:

is_injective(phi,pr[,c,s]); phi map, pr preimage ring, c int, s string

Return:

- 1 (type int) if phi is injective, 0 if not (if s is not given). - If s is given, return a list l of size 2, l[1] int, l[2] ring: l[1] is 1 if phi is injective, 0 if not l[2] is a ring with variables x(1),...,x(n),y(1),...,y(m) if the basering has n variables and the map m components, it contains the ideal 'ker', depending only on the y(i), the kernel of the given map

Note:

Three differnt algorithms are used depending on c = 1,2,3. If c is not given or c=0, a heuristically best method is chosen. The basering may be a quotient ring. However, if the preimage ring is a quotient ring, say pr = P/I, consider phi as a map from P and then the algorithm returns 1 if the kernel of phi is 0 mod I. To access to the ring l[2] and see ker you must give the ring a name, e.g. def S=l[2]; setring S; ker;

Display:

The above comment is displayed if printlevel >= 0 (default).

LIB "algebra.lib";
int p = printlevel;
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:
is_injective(phi,r);              // a,b,c ---> x-w,u2w+1,yz-v
==> 1
ring R = 0,(x,y,z),dp;
ideal i = x, y, x2-y3;
map phi = R,i;                    // a map from R to itself, z --> x2-y3
list l = is_injective(phi,R,"");
==> 
==> // The 2nd element of the list is a ring with variables x(1),...,x(n),
==> // y(1),...,y(m) if the basering has n variables and the map is
==> // F[1],...,F[m].
==> // It contains the ideal ker, the kernel of the given map y(i) --> F[i].
==> // To access to the ring and see ker you must give the ring a name,
==> // e.g.:
==>      def S = l[2]; setring S; ker;
==>         
l[1];
==> 0
def S = l[2]; setring S;
ker;
==> ker[1]=y(2)^3-y(1)^2+y(3)