D.4.1.11 finitenessTest

Procedure from library algebra.lib (see algebra_lib).

Usage:

finitenessTest(J[,v]); J ideal, v intvec (say v1,...,vr with vi>0)

Return:

a list l with l[1] integer, l[2], l[3], l[4] ideals - l[1] = 1 if var(v1),...,var(vr) are in l[2] and 0 else - l[2] (resp. l[3]) contains those variables which occur, (resp. occur not) as pure power in the leading term of one of the generators of J, - l[4] contains those J[i] for which the leading term is a pure power of a variable (which is then in l[2]) (default: v = [1,2,..,nvars(basering)])

Theory:

If J is a standard basis of an ideal generated by x_1 - f_1(y),..., x_n - f_n with y_j ordered lexicographically and y_j >> x_i, then, if y_i appears as pure power in the leading term of J[k]. J[k] defines an integral relation for y_i over the y_(i+1),... and the f’s. Moreover, in this situation, if l[2] = y_1,...,y_r, then K[y_1,...y_r] is finite over K[f_1..f_n]. If J contains furthermore polynomials h_j(y), then K[y_1,...y_z]/<h_j> is finite over K[f_1..f_n].

Example:

LIB "algebra.lib";
ring s = 0,(x,y,z,a,b,c),(lp(3),dp);
ideal i= a -(xy)^3+x2-z, b -y2-1, c -z3;
ideal j = a -(xy)^3+x2-z, b -y2-1, c -z3, xy;
finitenessTest(std(i),1..3);
→ [1]:
→    0
→ [2]:
→    _[1]=y
→    _[2]=z
→ [3]:
→    _[1]=x
→    _[2]=a
→    _[3]=b
→    _[4]=c
→ [4]:
→    _[1]=z3-c
→    _[2]=y2-b+1
finitenessTest(std(j),1..3);
→ [1]:
→    1
→ [2]:
→    _[1]=x
→    _[2]=y
→    _[3]=z
→ [3]:
→    _[1]=a
→    _[2]=b
→    _[3]=c
→ [4]:
→    _[1]=z3-c
→    _[2]=y2-b+1
→    _[3]=x2-z+a