Conclusion.

As a final conclusion of this section, all the results ([7,3,9]) give just sufficient conditions for the specialization of a Gröbner basis to be a Gröbner basis:

• The approach in [3] searches a set of ``good specializations" verifying this condition. Although from a practical point of view this is the most interesting question, it ``can lead to lengthy calculations which may be too much for some computer algebra systems", [3].
• In [7] and [9] we find a test to decide if a given specialization is good or not. The computations needed to realize these tests are efficient only in particular cases or when the specialization is such that no leading terms of the polynomials in the Gröbner basis vanishes. (In the other cases one more Gröbner basis computation is needed, which can be, in the worst case, the Gröbner basis of the specialization). Remark that the approach in [9] requires the computation of a Gröbner Basis over an integral domain which is usually harder that when the computations are done with polynomials with coefficients in a field.