Computing a set of good specialization parameter conditions.

Let J be a homogeneous ideal in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$, $G=\{g_1,\ldots,g_s\}$ its Gröbner basis with respect to a block $\underline{X}$-degree-compatible ordering $\sigma$ in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$. Every polynomial g_j has the structure

$$g_j(\underline{T},\underline{X})= \sum_{i=1}^{m_j} u_{i,j}(... ...j}>_{\overline{\sigma}}\alpha_{2,j}>_{\overline{\sigma}}\ldots$$

with $\overline{\sigma}$ the restriction of $\sigma$ to ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{X}]$. For a list of s nonnegative integers $L=[n_1,\ldots,n_s]$ with $0\leq n_j\leq m_j$, the specialization condition defined by L is given by

$$u_{j,1}(\underline{T})=0,\ldots,u_{j,n_j-1}(\underline{T})=0, u_{j,n_j}(\underline{T})\neq 0,\qquad 1\leq j\leq s$$

By using corollary 1, it is clear that, for any specialization verifying the conditions indicated by L, if the Hilbert Function of the monomial ideal generated by $\{{\bf X}^{\alpha_{j,n_i}}:1\leq j\leq s\}$ agrees with the Hilbert Function of the monomial ideal generated by $\{{\bf X}^{\alpha_{j,1}}:1\leq j\leq s\}$ then the specialization of G is a Gröbner Basis of the specialization of J (with respect to $\overline{\sigma}$). This consequence of corollary 1 provides a nice way of computing a set of conditions which are good for specializing the Gröbner Basis of J already computed in ${\rm I\kern -2.2pt K\hskip 1pt}[\underline{T},\underline{X}]$: i.e. those whose Hilbert function computed with the leading terms already present agrees with the ``generic" Hilbert function. In order to avoid the consideration of all the possible cases ( $(m_1+1)\cdot \ldots\cdot (m_s+1)$) the following lemma and its remarks show how to know in advance that such a condition list is bad once one such list has been previously classified as bad.

Lemma 1   Let $\geq$ be the ordering introduced in the set of conditions lists for G defined by

$$[n_1,\ldots,n_s]\geq[k_1,\ldots,k_s] \quad\Longleftrightarrow\quad n_1\geq k_1,\ldots,n_s\geq k_s.$$

If L is a condition list for G bad for specialization then every condition list L' for G with L<L' is also bad for specialization

First, it is clear that the list ${\bf0}=[0,\ldots,0]$ provides a good specialization condition. Next, it is analyzed the case of the vanishing of some $\underline{T}$-coefficients into only one g_j polynomial: for every $j\in\{1,\ldots,s\}$ the biggest (for $\geq$) conditions lists for G with only one non zero element which are good for specialization. Once these lists have been determined, we know, by the previous lemma that we do not need to consider any conditions list for G strictly bigger than one of such lists. Proceeding in a similar way with the case of only two, three, etc non-zero elements and with conditions lists not strictly bigger than those already located in the previous step, we exhaust all the possibilities.

Graphically this means that, for this Gröbner Basis,

$$\begin{array}{cccccc} g_1(\underline{T},\underline{X}) \right... ...nderline{T}) & \ldots & u_{sm_s}(\underline{T})\\ \end{array}$$

if we consider the set ${\cal A}$ of all lists $[n_1,\ldots,n_s]$ with $0\leq n_j\leq m_j$ (the different specialization conditions), we are looking for the maximal elements of the subset:

$${\cal S}=\{[n_1,\ldots,n_s]\in{\cal A}:[n_1,\ldots,n_s]\;\; \hbox{good for specialization}\}$$

with respect to the order relation $\geq$ in ${\cal A}$ defined in Lemma 1. If $L_1,\ldots,L_k$ are these maximal elements of ${\cal S}$we have obtained that if, for some i, $L=[n_1,\ldots,n_s]\leq L_i$ then the verification of the conditions

$$u_{j,1}(\underline{T})=0,\ldots,u_{j,n_j-1}(\underline{T})=0, u_{j,n_j}(\underline{T})\neq 0,\qquad 1\leq j\leq s$$ (2)

implies that the specialization of the considered Gröbner Basis G gives a Gröbner Basis of the specialized ideal.

In conclusion, every maximal element $L_i=[k_1,\ldots,k_s]$ of ${\cal S}$provides a set of $(k_1+1)\cdot\ldots\cdot(k_s+1)$ conditions with each one having the structure showed in 2. This procedure, as it has been explained taking only into account the set ${\cal A}$, can be easily optimized if we take also into account the structure of the polynomials $g_j(\underline{T},\underline{X})$'s: for example, if it is known that the coefficient $u_{ji}(\underline{T})$ is a non zero constant then we can discard directly from the set ${\cal A}$ those lists whose j-th element is bigger than i.