Singular Manual: finitenessTest

  | #### D.4.2.11 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. do not occur) 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]. For a proof cf. Prop. 3.1.5, p. 214. in [G.-M. Greuel, G. Pfister: A SINGULAR Introduction to Commutative Algebra, 2nd Edition, Springer Verlag (2007)] 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

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