Gröbner bases under specializations.

This section is devoted to present the state of the art in this kind of techniques. Let ${\rm I\kern -2.2pt K\hskip 1pt}$ be a field and $\underline{X}=(X_1,\dots,X_n)$, $\underline{T}=(T_1,\dots,T_m)$ sets of variables. We call specialization any ring homomorphism $\varphi:{\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]\rightarrow {\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]$ such that $\varphi({\rm I\kern -2.2pt K\hskip 1pt}[\underline{T}])={\rm I\kern -2.2pt K\hskip 1pt}$ and $\varphi(X_i)=X_i, i=1,\dots,n$. The specialization associated to a point $t^{(0)}\in{\rm I\kern -2.2pt K\hskip 1pt}^m$ is the one defined by $\varphi^{(0)}(p(\underline{T},\underline{X})):=p(t^{(0)},X)$, for $p(\underline{T},\underline{X})\in{\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$. Let $I=<F_1(\underline{T},\underline{X}),\dots,F_m(\underline{T}, \underline{X})>\subseteq {\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ be an ideal, > a monomial ordering in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$ and $G=\{g_1(\underline{T},\underline{X}),\dots,g_s(\underline{T},\underline{X})\}$ a Gröbner basis of I with respect to >. The most practical question that we would like to solve is to determine the subset W of ${\rm I\kern -2.2pt K\hskip 1pt}^m$ such that $t^{(0)}\in W$ if and only if its associated specialization $\varphi^{(0)}$ verifies that $\varphi^{(0)}(G)$ is a Gröbner basis of $\varphi^{(0)}(I)$. From a computational point of view this is the most interesting aspect, as once we have computed W, every time that we are given a value to the parameters we just have to test if it belongs or not to W to determine if we have or not a Gröbner basis. There are two possibilities to deal with this problem: to perform the initial Gröbner Bases computation in ${\rm I\kern -2.2pt K\hskip 1pt}(\underline{T})[\underline{X}]$ or in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$.