Simplification of the set of conditions.

As described in the previous section, we have obtained a set of good specialization conditions. Each condition different from the first one, ie that corresponding to the list $[0,\ldots,0]$, is presented in the following way:

$$p_1(\underline{T})=0,\ldots,p_k(\underline{T})=0, q_1(\underline{T})\neq 0,\ldots,q_m(\underline{T})\neq 0$$ (3)

where $k\geq 1$, $m\geq 1$ and the p_i's and the q_i's are polynomials in ${\mathchoice {\hbox{$\textstyle {\rm Z\!\!Z}$ }} {\hbox{$\textstyle {\rm Z\!\!... ...e {\rm Z\!\!Z}$ }} {\hbox{$\scriptscriptstyle {\rm Z\!\!Z}$ }}}[\underline{T}]$ non constant. In what follows it is showed how the description of every condition can be simplified.

In the case of one parameter, $\underline{T}=t$, if

$$p(t)=\gcd(p_1(t),\ldots,p_k(t))$$

the considered condition (3), with $\underline{T}=t$, is equivalent to

$$p(t)=0, q_1(t)\neq 0,\ldots,q_m(t)\neq 0$$ (4)

If p(t)=1, ie the p_i's are coprime, then this condition is directly discarded since it would be never verified. If $\deg(p(t))\geq 1$ then by defining iteratively

$$Q_{0}(t)=p(t),\qquad Q_{i+1}(t)=\frac{Q_i(t)}{\gcd(Q_i(t),q_i(t))},\quad\hbox{if} \;\; i\geq 0$$

it is easy to verify that the condition (4) is equivalent to

 

Q_m+1(t)=0 (5)

In the case of several parameters, it is clear that the same approach can not be further followed but several heuristics can be adopted in order to simplify the output:

• To check for emptyness with respect to the equalities: if the Grobner Basis of $p_1(\underline{T}),\ldots,p_k(\underline{T})$ with respect to any monomial ordering is equal to 1 then the corresponding condition is discarded.
• To apply a ``lazy factorization" approach as in the GPT strategy (see [6] or subsection 1.4).