Gianni's approach.

Given a specialization $\varphi$, we will say that an ordering > on the monomials of ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ is $\varphi$-admissible if it is a block ordering with respect to $\underline{T}$ and $\underline{X}$, and $\underline{X} > \underline{T}$, i.e., if >₁ (resp. >₂) is the restriction of > to the $\underline{X}$-variables (resp. to the $\underline{T}$-variables), then:

$$\bf {T}^B\bf {X}^A > \bf {T}^D\bf {X}^C \Longleftrightarrow ... ... \bf {X}^A =\bf {X}^C \hbox{ and }\bf {T}^B >_2 \bf {T}^D\cr}$$

for all $A,C\in{\rm I\kern -2.1pt N\hskip 1pt}^n$ and for all $B,D\in{\rm I\kern -2.1pt N\hskip 1pt}^m$. For a subset H of ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$, Lt(H) denotes the ideal generated by $\{lt(f)/f\in H\}$, where lt(f) denotes the leading term of f. If $\varphi$ is a specialization then $lt_{\varphi}(f)$ ( $f \in{\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$) will denote $U(\underline{T}){\bf X}^A$where ${\bf X}^A$ is the biggest $\underline{X}$-monomial in f. Thus $Lt_{\varphi}(H)$ will denote the ideal generated by $\{lt_{\varphi}(f)/f\in H\}$.

Theorem 1 ([7])   Let $I\subset{\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ be an ideal, $\varphi$ a specialization and G a Gröbner basis for I with respect to a $\varphi$-admissible ordering. If

$$Lt(\varphi(I))=\varphi(Lt_{\varphi}(I))$$ (1)

then $\varphi(G)$ is a Gröbner basis for $\varphi(I)$

Remark that, as G is a Gröbner basis of I, we can write

$$Lt(I)=<lt(g_1),\dots,lt(g_s)>$$

, and, therefore, we know the generators of the ideal to the right of the equality (1), $\varphi(Lt_{\varphi}(I))$:

$$\varphi(Lt_{\varphi}(I))=<\varphi(lt_{\varphi}(g_1)),\dots, \varphi(lt_{\varphi}(g_s))>.$$

However, we do not have a finite representation of the ideal to the left $Lt(\varphi(I))$ -- unless we know a priori a Gröbner basis of $\varphi(I)$ -- but finding this is the purpose of our computations. In the zero-dimensional and univariate case ( $\dim(\langle F_1,\ldots,F_m\rangle)=0$ and n=1) the results here described can be stated in a very practical way, avoiding any test.

Theorem 2 ([7])   Let $I\subset{\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ be a zero dimensional ideal and n=1, i.e. $\underline{X}=(X_1)$. Let $\varphi$ be a specialization. If G is a Gröbner basis for I with respect to a $\varphi$-admissible ordering then $\varphi(G)$ is a Gröbner basis for $\varphi(I)$.