Corollary 1
Let
J be a homogeneous ideal in
![${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$](img27.gif)
,
G its Gröbner basis with respect to
a block
![$\underline{X}$](img11.gif)
-degree-compatible ordering
![$\sigma$](img97.gif)
in
![${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$](img27.gif)
and
![$\phi$](img100.gif)
the specialization map giving
values to the parameters in
![$\underline{T}$](img7.gif)
.
Let
![$\overline{\sigma}$](img101.gif)
be
the restriction of
![$\sigma$](img97.gif)
to
![${\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]$](img96.gif)
,
![$H_{Lt_{\overline{\sigma}}(G)}$](img102.gif)
the Hilbert function of the ideal
generated by the
![$\underline{X}$](img11.gif)
-leading power products of
G and
![$H_{Lt_{\overline{\sigma}}(\phi(G))}$](img103.gif)
the Hilbert function of the ideal
generated by the leading power products of
![$\phi(G)$](img104.gif)
.
If
then
![$\phi(G)$](img104.gif)
is a Gröbner basis of
![$\phi(J)$](img106.gif)
with respect to
![$\overline{\sigma}$](img101.gif)
.