# Solving a system of complex non-linear equations

I have a set of five equations which can be described as follows:

$m_{i}=\frac{k_{1}}{(x+a)^{i}} + \frac{k_{2}}{(b+d)^{i}}+ \frac{k_{3}}{c^{i}}$

for i=1 to 5 where \eqalign{ k_{1}&=\frac{a(x+a-b-c-d)+c(b+d)-xd}{(x+a-b-d)(x+a-c)} \\ } \eqalign{ k_{2}&=\frac{x(d-c)}{(x+a-b-d)(b+d-c)} \\ } \eqalign{ k_{3}&=\frac{xb}{(b+d-c)(x+a-c)} }

$m_{i}$, $i$=1 to 5 are constants and $x,a,b,c,d$ are variables.

I want to find $x,a,b,c,d$ satisfying these equations.

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This looks like a hard question, unless the symmetry makes it obvious or something already well-studied. To help people trying to help with this, it might help to know: Where do these equations come from and what do you need the answer for? Do you have actual numbers for the m_i, or do you want parametric solutions in terms of them? If you have actual numbers, is a numerical approximation to the solutions (say from a homotopy continuation method) sufficient? Do you want all the solutions or just one? Do you know anything about Groebner basis methods? How good a computer do you have? – Alexander Woo Mar 13 '12 at 18:40