System of quadratic complex equations I want to solve this system of N  non-linear equations without using a numerical method:
$x_{k}^{2}= \alpha_{k }+ \sum\limits_{m=1}^{N} (\beta_{km} x_{m} + \psi_{km} x_{m}^{*})$
With
$\left| x_{k}-1 \right| \leq 1 $
All values are complex and $x^{*} $ is the conjugate of x.  Any idea? An approximated solution could be usefull
 A: Put $X=[x_1,\cdots,x_N]^T$. We consider an equation in the form $X\circ X+A'+B'X+C'\overline{X}=0$ where $\circ$ is the Hadamard product, $A'$ is a vector and $B',C'$ are square matrices.
Step 1. Put, for every $k$,  $x_k=1+y_k$. Then $Y$ satisfies an equation in the previous form. Then we may assume that $|x_k|\leq 1$.
Step 2. Separate real and imaginary parts with $X=U+iV$. We obtain a system (in the unknown REAL vectors $U,V$) in the following form $\{f(U,V)=U\circ U-V\circ V+A+BU+CV=0,\;\;g(U,V)=2U\circ V+D+EU+FV=0\}$. 
Solving (exactly) this system is hopeless. For $N=2$, there are $2^4$ solutions in $\mathbb{C}$  (using the Grobner basis method, we must solve an irreducible  polynomial of degree $2^4$). I think that, more generally, there are $2^{2N}$ solutions in $\mathbb{C}$. Obviously, we keep only the real solutions!
You search local solutions ; then you can use the Newton's method. Let $F=[f,g]^T$. The recurrence formula is well-known: $[U_{n+1},V_{n+1}]^T=[U_n,V_n]^T-(DF([U_n,V_n]^T))^{-1}([f(U_n,V_n),g(U_n,V_n)]^T$. Here $DF([U,V]^T)=\begin{pmatrix}2\tilde{U}+B&-2\tilde{V}+C\\2\tilde{V}+E&2\tilde{U}+F\end{pmatrix}$ where $\tilde{U}=diag(u_1,\cdots,u_N)$ and $\tilde{V}=diag(v_1,\cdots,v_N)$
