#### D.4.2.4 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;

User manual for Singular version 4.4.1, 2025, generated by texi2html.