Working in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$.

The approaches to be considered here treat the specialization aspect for particular orderings (block or lexicographical orderings), particular ideals (zero dimensional and univariate ideals) and particular specializations. The most important difference between these approaches and the previous one is that some computations have to be made for every given specialization in order to check if the specialized Gröbner Basis is actually a Gröbner Basis of the specialized ideal. In the following we present the details of these two approaches.