What to do with a ``parametric" Gröbner Basis?

It is worth to remark that in many cases, when the methods considered here are applied, then other algorithms are to be applied too: for example once it is known a set of specializations (through a set of conditions into the parameters) such that the specialised Gröbner Basis for any of these values of the parameters is indeed a Gröbner Basis then one could ask, if for these specializations, if the considered ideal is zero dimensional ideal. Note moreover that in this case, since the leading monomials do not change after the considered specializations, we have also a basis of the quotient (involving only $\underline{X})$-monomials) which can be very useful to get several informations about the properties of the set of $\underline{X}$-zeros of the considered ideal when the parameters live inside the set determined by the set of considered conditions. If we consider the polynomials:

$$\matrix{ F_1=a x t^2 + b y z t -x(x^2+c y^2+d z^2)\cr F_2=a... ...c z^2+d x^2)\cr F_3=a z t^2 + b x y t -x(z^2+c x^2+d y^2)\cr},$$

with x,y,z the unknowns and a,b,c,d the parameters, first we compute the Gröbner Basis of the ideal generated by F₁, F₂ and F₃ in ${\mathchoice {\setbox 0=\hbox{$\displaystyle\rm Q$ }\hbox {\raise 0.15\ht0\hbo... ...\ht0\hbox to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}[a,b,c,d,x,y,z,t]$ with respect the grevlex (graded reverse lexicographical) ordering. It has 23 polynomials whose leading terms and monomials are:

• leading terms:

  [c,d,1,d²-c,c²-d,cd-1,c⁴d+cd⁴-2 c²d²-c³-d³+2 cd,c³+d³-3 cd+1,d²,cd-1,c²-d, cd-1,1,4 cd-d³-1+c⁴d+cd⁴-3 c²d²-c³,4cd-d³-1+c⁴d+cd⁴-3 c²d²-c³, c³-2 cd+1,d³-2 cd+1,1,4 cd-d³-1+c⁴d+cd⁴-3 c²d²-c³,cd³-c²d-d²+c, c²d²-2cd+1,c³d-c²-cd²+d,1 ]

• leading monomials:

  [x²z,x²y,x³,zy³,xy²z,xy³, y²z³,xyz³,x²z³,y⁴z,y⁴z,

  y⁵,x²y³,yz⁵,xz⁵,y³z³, xy²z³ ,x²yz³,z⁷,y²z⁵, y²z⁵,y²z⁵,

  x²z⁵]

The ``generic'' Hilbert function is in this case

$${\frac {{n}^{6}+3\,{n}^{5}+6\,{n}^{4}+7\,{n}^{3}+6\,{n}^{2}+3\,n+1}{ 1-n}}$$

and since, for example, the condition [c=0,d=1] (with any value for a and b) conveys to the same Hilbert Function, it is concluded that such condition is good for specialization. Nevertheless the condition [c=0,d=0] (with any value for a and b) conveys to the Hilbert Function

$${\frac {2\,{n}^{8}+5\,{n}^{7}+5\,{n}^{6}+{n}^{5}-5\,{n}^{4}-7\,{n}^{3} -6\,{n}^{2}-3\,n-1}{1-n}}$$

and thus is not good for specialization.

For the first case, we have computed a Grobner Basis that specializes correctly for any value of the parameters verifying [c=0,d=1]. This allows to get information about the solution set of the considered system by performing, for example, parametric computations into the quotient ring ${\mathchoice {\setbox 0=\hbox{$\displaystyle\rm Q$ }\hbox {\raise 0.15\ht0\hbo... ...box to0pt{\kern0.4\wd0\vrule height0.7\ht0\hss}\box0}}}[x,y,z,t]/<F_1,F_2,F_3>$ (no need of taking care about the values of a and b).