GPT Strategy.

The strategy, here denoted GPT, was introduced in [6] in order to avoid the growth of the coefficients when using Buchberger algorithm to compute a Gröbner Basis of a multivariate polynomial ideal. The main idea is to keep track, into the different steps of Buchberger algorithm, of a list of potential factors (presented in factored form) such that in many cases simplification is posible by dividing by the ``greatest between the common divisors that are easy to compute" and in this way to avoid gcd computations. In our case, it seems that to apply this strategy is even more natural than to apply it to avoid the growing of the integers as it was the initial motivation in [6]. In fact the list of potential factors, in our case and working in ${\rm I\kern -2.2pt K\hskip 1pt}(\underline{T})[\underline{X}$], is going to give us a set of open conditions on the parameters such that if they are verified it is clear that the specialization of the generic Gröbner Basis is going to give us a Gröbner Basis of the specialized ideal. The example in [6] is very clear about how to proceed: if the initial system is

$$g_0=ax^3+byz+c,\quad g_1=dxy+ez^2,\quad g_2=fxz+gy$$

then the list of open conditions is given by

$$\{a,d,f,g,c,b^2e^2,b^2e\}$$

and when all these polynomials in the parameters are different from 0, the specialization of the generic Gröbner Basis is a Gröbner Basis of the specialized ideal. In fact this is no more that a lazy application of the ``Dynamic Evaluation Method" of D. Duval (see [4]) but following only one open branch. The decision to make at this point is how much do we want to have simplified the list of open conditions. The case of only one parameter is especially suited to apply the so called GPT strategy since all the computations with the parameters reduce to compute gcd's of univariate polynomials which is not very expensive.