Help with computation and Gröbner basis

Question

I am learning a new software and a new topic (Gröbner basis). I have the following system of polynomial equations

$$\begin{align} 6-21(x_1x_2+x_1x_3+x_1x_4) &= 0 \\ 10-21(x_2x_1+x_2x_3+x_2x_4) &= 0 \\ 12-21(x_3x_1+x_3x_2+x_3x_4) &= 0 \\ 14-21(x_4x_1+x_4x_2+x_4x_3) &= 0 \end{align}$$

and I need to find the number of solutions. I have been told I need to use Gröbner basis. The problem is that I do not know for what ideal I am supposed to find the Gröbner basis.

Secondly, if someone has experience with Macaulay2, I would appreciate if he can walk me through how we input this. What I was thinking was QQ[x1,x2,x3,x4, MonomialTOrder =>Lex] to get a ring and then use lex. Then I= ideal( ) here my lack of understanding kicks in.

Thank you guys for the time and patience.

Answer 1

Entering

R=QQ[x_1,x_2,x_3,x_4,MonomialOrder=>Lex]
I=ideal(6-21*(x_1*x_2+x_1*x_3+x_1*x_4),10-21*(x_2*x_1+x_2*x_3+x_2*x_4),12-21*(x_3*x_1+x_3*x_2+x_3*x_4),14-21*(x_4*x_1+x_4*x_2+x_4*x_3))
gens gb I

in Macaulay2 yields

$(9261x_4^8+11907x_4^6+2119x_4^4-2083x_4^2-660,\\14868x_3+1713285x_4^7+1397088x_4^5-282715x_4^3-263654x_4,\\36344x_2-4806459x_4^7-3836700x_4^5+724215x_4^3+703040x_4,\\49560x_1+379701x_4^7-21168x_4^5-151261x_4^3+24432x_4)$.

The first generator is in $x_4$ only and set to $0$ has $8$ complex solutions, two real by wolfram alpha. The other generators have $x_i, i=1,2,3$ enter linearly so they are determined uniquely by $x_4$.

---

Added: For the actual solutions, add toString oo to the commands above, and get:

matrix {{9261*x_4^8+11907*x_4^6+2119*x_4^4-2083*x_4^2-660,
 14868*x_3+1713285*x_4^7+1397088*x_4^5-282715*x_4^3-263654*x_4,
 36344*x_2-4806459*x_4^7-3836700*x_4^5+724215*x_4^3+703040*x_4,
 49560*x_1+379701*x_4^7-21168*x_4^5-151261*x_4^3+24432*x_4}}

Then go to, say, maxima, setting $x_4^2=t$

allroots(ratsubst(t,x_4^2,9261*x_4^8+11907*x_4^6+2119*x_4^4-2083*x_4^2-660));

[t = 0.418936018521825, t = - 0.4422977258287621, t = - 0.5138272911813384, t = - 0.7485252872260103]

Then to take one of the eight $x_4=\pm \sqrt{t}$

x_4:sqrt(0.418936018521825);
float(solve(14868*x_3+1713285*x_4^7+1397088*x_4^5-282715*x_4^3-263654*x_4,x_3));
float(solve(36344*x_2-4806459*x_4^7-3836700*x_4^5+724215*x_4^3+703040*x_4,x_2));
float(solve(49560*x_1+379701*x_4^7-21168*x_4^5-151261*x_4^3+24432*x_4,x_1));
x_4:'x_4;

we get $(x_1,x_2,x_3,x_4) \approx (0.1924229778886889, 0.3620745025257184, 0.4754969835800162, 0.6472526697680164)$

The other seven are:

$(-0.1924229778886889, -0.3620745025257184, -0.4754969835800162, -0.6472526697680164)$,

$(-0.7291921856328131 i, -0.8790682559168096 i, 0.6058365989819648 i, 0.6650546788263069 i)$,

$(0.7291921856328131 i, 0.8790682559168096 i, -0.6058365989819648 i, -0.6650546788263069 i)$,

$(-0.6516600637949471 i, 0.5916424877796417 i, -0.8700192754156101 i, 0.7168174741043486 i)$,

$(0.6516600637949471 i, -0.5916424877796417 i, 0.8700192754156101 i, -0.7168174741043486 i)$,

$(0.5839194982220037 i, -0.6443776225155836 i, -0.7101001755028209 i, 0.8651735590192353 i)$,

$(-0.5839194982220037 i, 0.6443776225155836 i, 0.7101001755028209 i, -0.8651735590192353 i)$