D.4.1.4 algDependent

Procedure from library algebra.lib (see algebra_lib).

Usage:

algDependent(f[,c]); f ideal (say, f = f1,...,fm), c integer

Return:

a list l of size 2, l[1] integer, l[2] ring: - l[1] = 1 if f1,...,fm are algebraic dependent, 0 if not - l[2] is a ring with variables x(1),...,x(n),y(1),...,y(m) if the basering has n variables. It contains the ideal ’ker’, depending only on the y(i) and generating the algebraic relations between the f[i], i.e. substituting y(i) by fi yields 0. Of course, ker is nothing but the kernel of the ring map K[y(1),...,y(m)] —> basering, y(i) –> fi.

Note:

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. The basering may be a quotient ring.
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).

Example:

LIB "algebra.lib";
int p = printlevel; printlevel = 1;
ring R = 0,(x,y,z,u,v,w),dp;
ideal I = xyzu2w-1yzu2w2+u4w2-1xu2vw+u2vw2+xyz-1yzw+2u2w-1xv+vw+2,
x-w, u2w+1, yz-v;
list l = algDependent(I);
→ 
→ // The 2nd element of the list l is a ring with variables x(1),...,x(n),
→ // and y(1),...,y(m) if the basering has n variables and if the ideal
→ // is f[1],...,f[m]. The ring contains the ideal ker which depends only
→ // on the y(i) and generates the relations between the f[i].
→ // I.e. substituting y(i) by f[i] yields 0.
→ // 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];
→ 1
def S = l[2]; setring S;
ker;
→ ker[1]=y(2)*y(3)*y(4)+y(3)^2-y(1)+1
printlevel = p;