# Solving a multivariate polynomial system takes too long

I have tried to solve this system

Solve[2 w2 x2 + 2 w3 x3 - y1 - 4 w2^2 y1 - 4 w3^2 y1 + 2 w2 y2 + 2 w3 y3 +
4 w1 (w2 (x2 + y2) + w3 (1 + x3 + y3)) + 2 x2 z1 + 2 y2 z1 -
4 y1 z2 + 4 x2 z1 z2 + 4 y2 z1 z2 - 4 y1 z2^2 + 4 z1 z3 +
4 x3 z1 z3 + 4 y3 z1 z3 - 4 y1 z3^2 -
x1 (3 + 4 w2^2 + 4 w3^2 + 4 z2 + 4 z2^2 + 4 z3^2) == 0 && x2 y1 + y1 y2 - 4 z1 - 8 w2^2 z1 - 8 w3^2 z1 - 2 x2^2 z1 - 4 x3 z1 -
2 x3^2 z1 - 4 x2 y2 z1 - 2 y2^2 z1 - 4 y3 z1 - 4 x3 y3 z1 -
2 y3^2 z1 + 4 w2 z2 + 2 x2 y1 z2 + 2 y1 y2 z2 + 4 w3 z3 + 2 y1 z3 +
2 x3 y1 z3 + 2 y1 y3 z3 + 4 w1 (w2 + 2 w2 z2 + 2 w3 z3) +
x1 (x2 + y2 + 2 x2 z2 + 2 y2 z2 + 2 (1 + x3 + y3) z3) == 0 && -2 w2 x1 x2 + x2^2 + x3 + x3^2 - 2 w2 x2 y1 - 2 w2 x1 y2 + 2 x2 y2 -
2 w2 y1 y2 + y2^2 + y3 + 2 x3 y3 + y3^2 - 4 w2 z1 + 2 z2 -
8 w2 z1 z2 + 4 z2^2 + 4 z3^2 -
2 w3 (x1 (1 + x3 + y3) + y1 (1 + x3 + y3) + 4 z1 z3) +
2 w1 (3 + x2^2 + x3^2 + 2 x2 y2 + y2^2 + 2 y3 + y3^2 +
2 x3 (1 + y3) + 4 z2 + 4 z2^2 + 4 z3^2) == 0,
{x1, y1, z1, w1, x2, y2, z2, w2, x3, y3, z3, w3},Reals]


but there is no output after a lot of hours, even days; Mathematica just says "running". I do not have experience with Mathematica but I think that my code is correct (I want the solutions of this system for the real variables {x1, y1, z1, w1, x2, y2, z2, w2, x3, y3, z3, w3}). Can anybody help me? Is there something wrong with my code? I think that the only possible solution is x1=y1=z1=w1=x2=...=w3=0.

• Change Solve to FindInstance and you get {{x1 -> 0, y1 -> 0, z1 -> 0, w1 -> 0, x2 -> 0, y2 -> 0, z2 -> 0, w2 -> 0, x3 -> 0, y3 -> 0, z3 -> 0, w3 -> 0}} instantly. If you add , 2 after Reals to try and get another solution but it will also hang up. For numerics, list your equations in eqns instead of using && and then minimize the total of the square lhs of all equations with NMinimize[Total[eqns[[All, 1]]^2], Variables[eqns[[All, 1]]], Method -> {"RandomSearch", "RandomSeed" -> 1}]. The first element of the result should be very small. Experiment with different seeds. – flinty Oct 14 '20 at 21:17
• With the above numerical approach I was able to get extremely close solutions using a higher WorkingPrecision - for example try: {w1 -> -0.11603984606102981984, w2 -> -0.048742660701450897572, w3 -> -0.15398251942165984864, x1 -> 0.040630686206066148672, x2 -> -0.39544569948625734660, x3 -> 0.14526644155234218870, y1 -> 0.26347836001151220795, y2 -> -0.60809681960239221756, y3 -> -0.68146214684674298984, z1 -> -0.049324673997652289876, z2 -> -0.047880851383080449072, z3 -> -0.087246627145441448400} – flinty Oct 14 '20 at 21:19
• You consider 3 equations and 12 unknowns. There are several possible solutions ! – Ulrich Neumann Oct 14 '20 at 21:35