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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:
 


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