Simultaneous Equations

[Diophantus]

I am trying to find the solutions to the following system over the complex numbers:


$$x^4 + y^4 + z^4 +4xyz - 4x^2 - 4y^2 - 4z^2 +5 =0$$
$$4x^3 + 4yz - 8x =0$$
$$4y^3 + 4xz - 8y =0$$
$$4z^3 + 4xy - 8z =0$$


I calculated the resultant:

$$xyz - 2x^2 - 2y^2 - 2z^2 +5$$

which would give a necessary condition for solutions but I've been messing about for hours and I still can't see how to proceed.

Any suggestions?

 

[robphy]

Can you take advantage of the symmetry among the variables?

 

[mathman]

You have 4 equations with 3 unknowns. In general there won't be a solution (although your particular equations could be solvable).

Also the third equation doesn't look right, shouldn't the middle term be 4xz?

 

[robphy]

Seems like x=y=z=1 is a solution.

 

[Diophantus]

Also the third equation doesn't look right, shouldn't the middle term be 4xz?

Indeed you are right.

So the symmetry implies that if (a,b,c) is asloution then so are (a,c,b) , (b,a,c) , (b,c,a) , (c,a,b) , and (c,b,a).

I'm really looking for a way of obtaining the whole solution set. I have done similar problems in the past by getting a resultant doing some substitution in order to get a list of candidates (containing all solutions) which can then be chencked.

 

[robphy]

It appears that your last three equations are partial derivatives of the first equation.
I suspect a geometric interpretation might help guide you to the solution.

 

[Diophantus]

Sorry I should have been more honest from the outset. I'm trying to find the singular points of the surface defined in the top equation. These points are exactly the points on the surface which satisfy the equations of the derivatives. I didn't say that before because I thought it would confuse matters. In effect the geometric problem was the hard problem which was supposed to be made easier by translating it into an algerbaic one.

 

[Hurkyl]

Are you allowed to ask magma?


Your computation of resultant is unfamiliar to me -- the one I know takes two polynomials and a variable as input, and outputs a single polynomial that does not contain the given variable. So I don't know what you computed.

 

[Diophantus]

The 'resultant' I was using is not technically the same as the usual definition I believe. What I was doing was setting:

$$4f - f_x - f_y - f_z = 0$$

which gives a necessary condition for the solutions of the four equations.

Not familiar with magma I'm afraid. The important thing is that I really need to be able to know how to do it but if you could generate the solution set using magma then I guess it would be, helpful in the sense of knowing what we are looking for.

 

[robphy]

Maple:

> eq:={x^4+y^4+z^4+4*x*y*z-4*x^2-4*y^2-4*z^2+5=0, 4*x^3+4*y*z-8*x=0, 4*y^3+4*z*x-8*y=0, 4*z^3+4*x*y-8*z=0};

          4    4    4                2      2      2
  eq := {x  + y  + z  + 4 x y z - 4 x  - 4 y  - 4 z  + 5 = 0,

           3                       3
        4 x  + 4 y z - 8 x = 0, 4 y  + 4 z x - 8 y = 0,

           3
        4 z  + 4 x y - 8 z = 0}

> solve(eq,{x,y,z});

  {z = 1, y = 1, x = 1}, {z = -1, y = 1, x = -1},

        {z = -1, x = 1, y = -1}, {z = 1, x = -1, y = -1}

The surfaces [ plotted with implicitplot3d ] look interesting... but weren't immediately suggestive of a strategy... with the exception of the obvious symmetry in the variables.

 

[Diophantus]

Thanks robphy, I'll ponder it a bit longer and report my progess later.

 

[Diophantus]

I sustituted my 'resultant' into the first equation to get:

$$(x^2 + 5)(x^2 - 1) + (y^2 + 5)(y^2 - 1) + (z^2 + 5)(z^2 - 1) = 0$$

which looks quite nice but I can't see any immediate use.

I also used the 'resultant' to get:

$$x = \frac{yz}{4} \pm \frac{1}{4}\sqrt{40 - 16x^2 - 16z^2 + y^2 z^2}$$

Alas though substituing into any of the original equations just gives a mess.

 

[Diophantus]

I'm still stuck. Anyone?