# Mathematica can't solve this system of equations

I want to solve this system but I can't

Solve[(1 - x^2) (-W - 3 x + Sqrt[6] Sqrt[z])
==
-W^4 x (-1 + x^2) y +
3 W y (W^3 (-1 + x^2) (-1 + Sqrt[1 - x^2] y) -
6 Sqrt[6] (y + Sqrt[1 - x^2] (-1 + z))^2 Sqrt[z] +
W^3 (-1 + x^2) z)
==
9 Sqrt[6] (y + Sqrt[1 - x^2] (-1 + z))^2 (1 - 2 z) Sqrt[z]
-1/3 W^3 (-1 + x^2) z (Sqrt[6] W x
- 9 (-1 + Sqrt[1 - x^2] y + z))
==
3 W (W^3 (-1 + x^2) (-1 + Sqrt[1 - x^2] y) -
6 Sqrt[6] (y + Sqrt[1 - x^2] (-1 + z))^2 Sqrt[z] +
W^3 (-1 + x^2) z)
== 0, {x, y, z, W}]

• it does not appear to be a system ie a list of equations but a single expression with consecutive Equal signs.Consider breaking up the successive equalities into their corresponding parts ie transform x==y==z into {x==y,y==z,x==z} – user42582 Oct 26 '16 at 7:59
• @user42582 That's not a problem. It's valid syntax. (BTW it can be expanded using LogicalExpand.) I believe that reason why Solve is so slow that after eliminating all the square roots it ends up with a high order polynomial system of equations with complicated coefficients. It is simply too slow to work with such large expressions. This also means that the solution would look uselessly complicated, so it's not worth computing the exact symbolic result. – Szabolcs Oct 26 '16 at 8:52
• Evidence for this is also the high memory usage of the kernel when evaluating this Solve. On my machine it keeps hovering around 1 GB (going up and down) after running for a couple of minutes. – Szabolcs Oct 26 '16 at 8:54
• @Szabolcs you are right , when run this code about all of my RAM and a large part of my CPU used. I am waiting for about 1 hour with any result. – milad Oct 26 '16 at 9:55
• I suggest you use NSolve. It also took a long time but it finished and found 967 solutions. I don't know if this solution set is exhaustive. – Szabolcs Oct 26 '16 at 10:10

## 1 Answer

Making use ofFindRoot instead of Solve, one obtains by

FindRoot[{sys}, {{x, 0.1} , {y, 0.1}, {z, 0.1}, {W,0.1}}]


{x -> 0.57735, y -> 0.100001, z -> 0.5, W -> 9.21509*10^(-18)}.

Addition. Subsituting W->2 in the system and NSolving it in $x,y,z$, one obtains

NSolve::infsolns: Infinite solution set has dimension at least 1. Returning intersection of solutions with -((121484 x)/178835)-(113492 y)/178835+(171802 z)/178835 == 1.

and

{{x -> 1., y -> -2.64617, z -> 0.}, {x -> -1., y -> -0.505331, z -> 0.}, {x -> -1., y -> 0., z -> 0.33382}, {x -> -1., y -> 0., z -> 0.33382}, {x -> -1., y -> 0., z -> 0.33382}, {x -> 1., y -> 0., z -> 1.74805}, {x -> 1., y -> 0., z -> 1.74805}, {x -> 1., y -> 0., z -> 1.74805}}

Addition 2. By the changes Sqrt[z]==s and Sqrt[1-x^2]==t the system under consideration can be reduced to a polynomial system over Q(Sqrt[6]). The reduced system can be solved in terms of higher degree polynomials via Groebner on powerful comp. It is clear that solution would be huge and useless. I find it monkey business.

• There are at least 6 real solutions of the system under consideration. – user64494 Oct 26 '16 at 10:27
• Thanks, The "Find Root" find one solution but I think this system have more answers – milad Oct 26 '16 at 11:07
• @ milad : Do you really need all the ones? – user64494 Oct 26 '16 at 11:10
• Yes, I am studying dynamical system of a model in cosmology and I must check all answers for investigate this model is good or no – milad Oct 26 '16 at 11:34
• @ milad : Then how about the case when the solution set is infinite? – user64494 Oct 26 '16 at 11:37