I am interested in analytical solutions for a system of nonlinear equations.

(The question was first asked at math.SE, where (after 1months and one rounds of bounty) there is only interesting analysation that the problem corresponds to high-order polynomial. I'm hoping to get more insight here, in particular whether there are methods which can deal with such high-order polynomials or whether it is likely to be not possible to solve it).

**Motivation**: The source of the question is a very convinient method to create random matrices with special properties. Mathematica can give me solutions up to certain sizes of the linear system, but I would like to have it for arbitrary size N. I can also use numerical algorithms (which I am doing at the moment), but for N in the order of $N\approx10.000$, they are quite slow.

**System of nonlinear equations**:

$$ (w_i \cdot \sum_{j=1}^N w_j) - w_i^2 = d_i $$ for $i=1...N$, and $w$ and $d$ are vectors with $N$ dimensions, and $w_i$ and $d_i$ is the $i$-th component of the vector. Both $d_i$ and $w_i \in \mathbb{R_+}$. I am providing the vector $d$ (i.e. N real non-negative numbers), and want to solve for $w_i$.

How can $w_i$ be written down in a closed-form for a given vector $d$, for arbitrary N?

**Example:** If N=3 we have the following system of equations:

$$ w_1 \cdot (w_2 + w_3) = d_1 \\ w_2 \cdot (w_1 + w_3) = d_2 \\ w_3 \cdot (w_1 + w_2) = d_3 $$

with $w_i, d_i \in \mathbb{R}$. For a given vector $d=(d_1,d_2,d_3)$, I want to get $w=(w_1,w_2,w_3)$.

**Rewriting:** There is a nice way to rewrite the question, but I am not sure whether it actually helps:

Let's set $c=\sum_{j=1}^N w_j$, which is the sum of all weights. What we have now:

$$c \cdot w_i - w_i^2 = d_i \\ w_i^2 - c \cdot w_i + d_i = 0 $$ which has two solutions:

$$w_{i_{1,2}} = \frac{c}{2} \pm \sqrt{ \left(\frac{c}{2}\right)^2 - d_i} $$ and the normalisation constant $c$ can be calculated by the sum of all weights:

$$\sum_{j=1}^N w_j = \sum_{j=1}^N \left(\frac{c}{2} \pm \sqrt{ \left(\frac{c}{2}\right)^2 - d_j} \right) = c $$

Is there a closed-form solution for c?

**Analysation from math.SE:**
For completness, I copy the analysation of Robert Israel at math.SE:

$n=3$ is rather easy: $w_i$ satisfies a quadratic polynomial.

For $n=4$, each $w_i$ satisfies a rather nasty polynomial of degree $8$ (but involving only even powers). Thus there is a solution in terms of radicals, but it won't be pleasant.

For $n=5$, it seems each $w_i$ satisfies a polynomial of degree $22$. A solution in radicals is not to be expected. Thus with $d_1 = -7, d_2 = 3, d_3 = 9, d_4 = 7, d_5 = 8$, $w_5$ satisfies $$ 81\,{w_{{5}}}^{22}+7074\,{w_{{5}}}^{20}+198792\,{w_{{5}}}^{18}+2887764 \,{w_{{5}}}^{16}+23487600\,{w_{{5}}}^{14}+99587008\,{w_{{5}}}^{12}+ 69082752\,{w_{{5}}}^{10}-1402988992\,{w_{{5}}}^{8}-6995300352\,{w_{{5} }}^{6}-14191984640\,{w_{{5}}}^{4}-13095665664\,{w_{{5}}}^{2}- 4492099584 = 0 $$

EDIT: This is an irreducible polynomial of degree $11$ in $x = w_5^2$. Maple doesn't do Galois groups of polynomials of degree $11$, but GAP does, and confirms that its Galois group is $S_{11}$. In particular, there is no solution in radicals.