Strongly independent set of variables
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a strongly independent set of variables modulo the ideal generated
A set of variables S is strongly independent modulo an ideal I if no leading term of an element of the Gröbner basis of I consists entirely of elements of S. A set is maximally strongly independent if no proper superset of it is strongly independent. Two maximally strongly independent set may be of different size.
Gröbner bases in the format returned by
The following example has been given by Moeller and Mora in 1983.
G:=map([X0^8*X2, X0*X3, X1^8*X3, X1^7*X3^2, X1^6*X3^3, X1^5*X3^4, X1^4*X3^5, X1^3*X3^6, X1^2*X3^7, X1*X3^8], poly, [X3, X2, X1, X0]): groebner::stronglyIndependentSets(G)
The Gröbner basis of an ideal: a list.
List of the form
[d, S, M], where
an integer equal to the dimension of the ideal generated by
the greatest strongly independent set of variables, and
a set consisting of all maximal strongly independent sets of variables
or a piecewise consisting of such lists.
 Kredel H. and V. Weispfenning, “Computing dimension and independent sets for polynomial ideals”, JSC volume 6 (1988), 231-247.