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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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