First I describe my problem (note that I am not so familiar with algebraic geometry). I have the following family $\mathcal{F}$ of $19$ polynomials in the ring $\mathbb{C}[x_1,\dots,x_{10}]$, with complex parameters $p_1,p_2,p_3,p_4,p_\infty$:
C1 = -4 + p1^2 + p2^2 + p3^2 - p1*p2*x1 + x1^2 - p1*p3*x2 + x2^2 - p2*p3*x3 + x1*x2*x3 + x3^2 + p1*p2*p3*x7 - p3*x1*x7 - p2*x2*x7 - p1*x3*x7 + x7^2;
C2 = -4 + p2^2 + p3^2 + p4^2 - p2*p3*x3 + x3^2 - p2*p4*x5 + x5^2 - p3*p4*x6 + x3*x5*x6 + x6^2 + p2*p3*p4*x8 - p4*x3*x8 - p3*x5*x8 - p2*x6*x8 + x8^2;
C3 = -4 + p1^2 + p3^2 + p4^2 - p1*p3*x2 + x2^2 - p1*p4*x4 + x4^2 - p3*p4*x6 + x2*x4*x6 + x6^2 + p1*p3*p4*x9 - p4*x2*x9 - p3*x4*x9 - p1*x6*x9 + x9^2;
C4 = -4 + p1^2 + p2^2 + p4^2 - p1*p2*x1 + x1^2 + p1*p2*p4*x10 - p4*x1*x10 + x10^2 - p1*p4*x4 - p2*x10*x4 + x4^2 - p2*p4*x5 - p1*x10*x5 + x1*x4*x5 + x5^2;
C6 = -4 + p1^2 + x1^2 - p2*(-p2 + p1*x1) + x6^2 - p1*x6*x9 + x9^2 + x1*x9*x8 - x1*x6*pinf - p2*x9*pinf + pinf^2 + (p2*x6 - x8)*(-x8 + p1*pinf);
C7 = -4 + x1^2 + p3^2 - x1*p3*x7 + x7^2 + p4^2 - x1*p4*x10 + x10^2 - p3*p4*x6 + x7*x10*x6 + x6^2 + x1*p3*p4*pinf - x7*p4*pinf - p3*x10*pinf - x1*x6*pinf + pinf^2;
C8 = -4 + p2^2 + p3^2 - p2*p3*x3 + x3^2 + x4^2 - p2*x4*x10 + x10^2 - p3*x4*x9 + x3*x10*x9 + x9^2 + p2*p3*x4*pinf - x3*x4*pinf - p3*x10*pinf - p2*x9*pinf + pinf^2;
C9 = -4 + p1^2 + x3^2 - p1*x3*x7 + x7^2 + p4^2 - p1*p4*x4 + x4^2 - x3*p4*x8 + x7*x4*x8 + x8^2 + p1*x3*p4*pinf - x7*p4*pinf - x3*x4*pinf - p1*x8*pinf + pinf^2;
R1 = p1*p2*p3*p4 - x1*p3*p4 - p1*x3*p4 + x7*p4 - p2*p3*x4 + x3*x4 - x2*x5 + p3*x10 - p1*p2*x6 + x1*x6 + p2*x9 + p1*x8 - 2*pinf;
R2 = -(p1*p2*p4) + x1*p4 + p2*x4 + p1*x5 - 2*x10 + p1*x3*x6 - x7*x6 - x3*x9 - p1*p3*x8 + x2*x8 + p3*pinf;
R3 = -(p1*p4) + 2*x4 + x1*x5 - p2*x10 + x2*x6 + x1*x3*x6 - p2*x7*x6 - p3*x9 - x1*p3*x8 + x7*x8 + p2*p3*pinf - x3*pinf;
R4 = -(p1*p2*p3) + x1*p3 + p2*x2 + p1*x3 - 2*x7 + p2*x4*x6 - x10*x6 - p2*p4*x9 + x5*x9 - x4*x8 + p4*pinf;
R5 = -(p1*p2) + 2*x1 + x2*x3 - p3*x7 + x4*x5 - p4*x10 + x3*x4*x6 - x3*p4*x9 - p3*x4*x8 + x9*x8 + p3*p4*pinf - x6*pinf;
R6 = p2*p3*p4 - x3*p4 - x1*p3*x4 + x7*x4 - p3*x5 + p1*p3*x10 - x2*x10 - p2*x6 + x1*x9 + 2*x8 - p1*pinf;
R7 = p1*p3*p4 - x2*p4 - x1*x3*p4 + p2*x7*p4 - p3*x4 - x7*x5 + x3*x10 - p1*x6 + 2*x9 + x1*x8 - p2*pinf;
R8 = -(p2*p4) + x1*x4 + 2*x5 - p1*x10 + x3*x6 - x7*x9 - p3*x8 + x2*pinf;
R9 = p1*p3 - 2*x2 - x1*x3 + p2*x7 - x4*x6 + p4*x9 + x10*x8 - x5*pinf;
R10 = p2*p3 - x1*x2 - 2*x3 + p1*x7 - x1*x4*x6 - x5*x6 + p1*x10*x6 + x1*p4*x9 - x10*x9 + p4*x8 - p1*p4*pinf + x4*pinf;
R11 = -(p3*p4) + x2*x4 + x1*x3*x4 - p2*x7*x4 + x3*x5 - p1*x3*x10 + x7*x10 + 2*x6 - p1*x9 - p2*x8 + p1*p2*pinf - x1*pinf;
I am interested in the algebraic variety generated by the zero locus of the family $\mathcal{F}$. In particular I expect that the dimension of the variety is actually 4. In order to calculate the dimension of my variety with Macaulay2, I perform the following steps:
kk = QQ;
kkk = frac(kk[p1,p2,p3,p4,pinf]);
R = kkk[x1,x2,x3,x4,x5,x6,x7,x8,x9,x10];
I = ideal (C1,C2,C3,C4,C6,C7,C8,C9,R1,R2,R3,R4,R5,R6,R7,R8,R9,R10,R11);
X = spec (R/I)
dim X
and I get the correct dimension (and the computation is reasonably fast): $\dim X = 4$.
At this point I want to isolate $6$ polynomials, $f_1,\dots,f_6$ in $\mathcal{F}$, such that if I run the following piece of code:
J = ideal(f1,...,f6);
X = spec(R/J);
dim X;
I get again that the dimension of X is $4$.
The problem is that I am not able to find out such sub-family $f_1,\dots,f_6$. I tried random combinations of $6$ polynomials, but for every such combination the computation in Macaulay2 doesn't end.
Why it is easier for Macaulay2 to compute the dimension of spec(R/I) (where there are 19 polynomials) than to compute the dimension of spec(R/J)?
Is there any way to find out this family of $6$ polynomials?
[Update:] I can use the command mingens (as suggested by Jesko Hüttenhain) to get a minimal Gröbner basis:
gB = gb I;
mgB = mingens gB;
then gB has 32 elements and mgB has 15 elements. The following ideal:
J = ideal(mgB_(0,7),mgB_(0,8),R1,C6,C7);
is such that $\dim J = 5$. I need one more polynomial, but for every combination I tried (picking the 6th poly from $\mathcal{F}$) the computation doesn't end.