Using Hilbert Function.

Let J be an ideal in ${\rm I\kern -2.2pt K\hskip 1pt}[X_0, \ldots, X_n]= {\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]$. J is said homogeneous if the relation $F \in J$ implies that all the homogeneous components of F are in J. Let ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]_s$ denote the set of homogeneous polynomials of total degree s (including the zero polynomial) which is a ${\rm I\kern -2.2pt K\hskip 1pt}$-vector space of dimension $\theta(n, s)={{n+s} \choose{s}}$. If J is a homogeneous ideal, we let

$$J_s=J \cap {\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]_s.$$

Then the Hilbert function of J is defined by

$$H_J(s)= \dim_{{\rm I\kern -2.2pt K\hskip 1pt}} {\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]_s/J_s.$$

It is also possible to define the Hilbert Function for inhomogeneous ideals: it is enough to replace ${\rm I\kern -2.2pt K\hskip 1pt}[X_0, \ldots, X_n]_s$ by ${\rm I\kern -2.2pt K\hskip 1pt}[X_1, \ldots, X_n]_{\leq s}$, the set of polynomials in ${\rm I\kern -2.2pt K\hskip 1pt}[X_1, \ldots, X_n]$ of total degree at most s (including the zero polynomial). Following the ideas in [12] it is posible to use the Hilbert function, to check if the specialisation of a Gröbner Basis of a homogeneous (with respect to $\underline{X}=(X_0, \ldots, X_n)$) ideal Jin ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ with respect to a block ordering is a Gröbner Basis of the specialised homogeneous ideal. The main tool is the following theorem appearing in [12] and its corollary to the situation we are dealing with.

Theorem 3 ([12])   Let J be an ideal in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]$ and $\sigma$ a degree-compatible term ordering. Let $G=\{g_1,\ldots,g_k\}$ be a set of elements of J. If

$$H_{Lt_{\sigma}(J)}=H_{Lt_{\sigma}(G)}$$

then G is a Gröbner basis of J with respect to $\sigma$.

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

$$H_{Lt_{\overline{\sigma}}(G)}=H_{Lt_{\overline{\sigma}}(\phi(G))}$$

then $\phi(G)$ is a Gröbner basis of $\phi(J)$ with respect to $\overline{\sigma}$.

The key point here is to compute a ``generic" Gröbner Basis of the ideal J regarded in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ with respect to a degree-compatible monomial ordering and to use this Gröbner Basis to derive the ``generic" Hilbert function of J considered as ideal in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]$. For a given specialization of the parameters, the specialization of the ``generic" Gröbner Basis is a Gröbner Basis if the ``generic" Hilbert function agrees with the Hilbert function of the monomial ideal generated by the leading power products of the specialized Gröbner Basis. To remark finally that the Hilbert function of a monomial ideal is very easy to compute and thus it seems to be an immediate tool to use in order to check the correctness of the specialization process.