|
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:
-
- 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
|