Singular

D.4.2.2 module_containment

Procedure from library algebra.lib (see algebra_lib).

Usage:
module_containment(p,P,M[,k]); p poly, P ideal, M ideal, k int
P = P[1],...,P[n] generators of a subalgebra of the basering,
M = M[1],...,M[m] generators of a module over the subalgebra K[P]

Assume:
ncols(P) = nvars(basering), the P[i] are algebraically independent

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

Display:
the polynomial h(y(1),...,y(m),z(1),...,z(n)) if k=0, resp. a comment how to access the relation check if k=1, provided printlevel >= voice+1 (default).

Theory:
The ideal of algebraic relations of all the generators p1,...,pn, s1,...,st given by P and S is computed introducing new variables y(j), z(i) and the product order: x^a*y^b*z^c > x^d*y^e*z^f if x^a > x^d with respect to the lp ordering or else if z^c > z^f with respect to the dp ordering or else if y^b > y^e with respect to the lp ordering again. p reduces to a polynomial only in the y(j) and z(i), linear in the z(i) <=> p is contained in the module.

Example:
 LIB "algebra.lib"; int p = printlevel; printlevel = 1; ring R=0,(x,y,z),dp; ideal P = x2+y2,z2,x4+y4; //algebra generators ideal M = 1,x2z-1y2z,xyz,x3y-1xy3; //module generators poly p1= x10z3-x8y2z3+2x6y4z3-2x4y6z3+x2y8z3-y10z3+x6z4+3x4y2z4+3x2y4z4+y6z4; module_containment(p1,P,M); ==> // y(2)*z(2)*z(3)^2+z(1)^3*z(2)^2 ==> 1 poly p2=z; list l = module_containment(p2,P,M,1); ==> ==> // 'module_containment' created a ring as 2nd element of the list. The ==> // ring contains the polynomial check which defines the algebraic relatio\ n ==> // for p. 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]; ==> 0 def S = l[2]; setring S; check; ==> x(3) printlevel=p;