       Re: Solve, when I already know 1 solution

• To: mathgroup at smc.vnet.net
• Subject: [mg73114] [mg73114] Re: [mg73080] Solve, when I already know 1 solution
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Sat, 3 Feb 2007 05:04:24 -0500 (EST)
• References: <200702010851.DAA11372@smc.vnet.net>

```On 1 Feb 2007, at 09:51, DOD wrote:

> I am trying to solve the following system of equations for, X, Y, and
> Z:
>
>
>
> {X((16  X ^3 - 39  X ^2  Y + 5  X  Y ^2 + 8  Y ^3 +  ((19  X ^2 -
> 10 X  Y +
> 3  Y ^2) )  Z + 3   ((X + Y) )  Z ^2 + Z ^3) ) == 9  c, X Y ((29  X
> ^2 - 2
> Y   ((4  Y + Z) ) - 2  X   ((9  Y + 5  Z) )) ) ==  (-9 )  c, Z
> ((10  X ^3 +
> 81  Y (Y - Z) ( (-1 ) + Z) +  X  Y   ((81 - &9  Y + 2  Z) ) + X
> ^2   (( (-
> 81 ) + 72  Y + 4  Z))) ) == 9  c}
>
> (sorry for the bad formatting, I don't know a convenient way to
> copy this
> info from mathematica.  Anyway, the precise polynomial isn't
> important. )
>
> I would like a symbolic solution in c.  So Solve chews on this forever
> without giving a solution.  Now, it so happens that I already know a
> solution is X=Y=Z =c^(1/4), and I strongly suspect that all the other
> solutions, real or imaginary, are of the form X=q c^(1/4),Y=r c^
> (1/4),Z= s
> c^(1/4), for some (q,r,s).  Is there any way I can use this info to
> help
> Solve along, and give me all the solutions?
>
>
> -- DOD
>

I doubt that in the case of a multivariate system knowing one
solution will be of any help in finding the general solution.
Moreover, I can't understand your conjecture that

> all the other
> solutions, real or imaginary, are of the form X=q c^(1/4),Y=r c^
> (1/4),Z= s
> c^(1/4), for some (q,r,s)

Given a non-zero c, any three number X,Y,Z can be written in the
above form for some (q,r,s). There are actually 4 obvious solutions
of this form, where q=r=s is an element of the set {I,-I,1,-1} but
these are certainly not all solutions (since for numerical values of
d it is easy to find solutions not equal to any of these).

Andrzej Kozlowski

```

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