#### D.4.2.1 algebra_containment

Procedure from library `algebra.lib` (see algebra_lib).

Usage:
  algebra_containment(p,A[,k]); p poly, A ideal, k integer.
  A = A[1],...,A[m] generators of subalgebra of the basering

Return:

               - k=0 (or if k is not given) an integer:
                 1  : if p is contained in the subalgebra K[A[1],...,A[m]]
                 0  : if p is not contained in K[A[1],...,A[m]]
               - k=1 : a list, say l, of size 2, l[1] integer, l[2] ring, satisfying
                 l[1]=1 if p is in the subalgebra K[A[1],...,A[m]] and then the ring
                 l[2]: ring, contains poly check = h(y(1),...,y(m)) if p=h(A[1],...,A[m])
                 l[1]=0 if p is not in the subalgebra K[A[1],...,A[m]] and then
                 l[2] contains the poly check = h(x,y(1),...,y(m)) if p satisfies
                 the nonlinear relation p = h(x,A[1],...,A[m]) where
                 x = x(1),...,x(n) denote the variables of the basering

Display:
  if k=0 and printlevel >= voice+1 (default) display the polynomial check

Note:
  The proc inSubring uses a different algorithm which is sometimes faster.

Theory:
  The ideal of algebraic relations of the algebra generators A[1],..., A[m] is computed introducing new variables y(i) and the product order with x(i) >> y(i).
  p reduces to a polynomial only in the y(i) <=> p is contained in the subring generated by the polynomials A[1],...,A[m].

Example:

    LIB "algebra.lib";
    int p = printlevel; printlevel = 1;
    ring R = 0,(x,y,z),dp;
    ideal A=x2+y2,z2,x4+y4,1,x2z-1y2z,xyz,x3y-1xy3;
    poly p1=z;
    poly p2=
    x10z3-x8y2z3+2x6y4z3-2x4y6z3+x2y8z3-y10z3+x6z4+3x4y2z4+3x2y4z4+y6z4;
    algebra_containment(p1,A);
    ==> // x(3)
    ==> 0
    algebra_containment(p2,A);
    ==> // y(1)*y(2)*y(5)^2+y(3)*y(5)^3+4*y(1)*y(2)*y(6)^2+4*y(6)^3*y(7)+2*y(2)*y\
       (5)*y(7)^2
    ==> 1
    list L = algebra_containment(p2,A,1);
    ==>
    ==> // 'algebra_containment' created a ring as 2nd element of the list.
    ==> // The ring contains the polynomial check which defines the algebraic rel\
       ation.
    ==> // To access to the ring and see check you must give the ring a name,
    ==> // e.g.:
    ==>                def S = l[2]; setring S; check;
    ==>
    L[1];
    ==> 1
    def S = L[2]; setring S;
    check;
    ==> y(1)*y(2)*y(5)^2+y(3)*y(5)^3+4*y(1)*y(2)*y(6)^2+4*y(6)^3*y(7)+2*y(2)*y(5)\
       *y(7)^2
    printlevel = p;

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