Decouple a system of quadratic equations

Question

We have $n$ real variables, $x_1, \ldots, x_n$. We'd like to solve the following system of quadratic equations. \begin{cases} \displaystyle\phantom{_{+1}}x_1 \sum_{j=1}^n x_j z_j = f_1(x_1);\\ \displaystyle \phantom{_{+1}}x_2 \sum_{j=1}^n x_j z_j = f_2(x_2);\\ \qquad\;\,\dots\\ \displaystyle \phantom{_{+1}}x_n \sum_{j=1}^n x_j z_j = f_n(x_n).\\ \end{cases} Here $f_i(x_i)$ are some quadratic functions in one variable. Is there an obvious way to decouple this into $n$ equations each of which depend on only one variable?

Answer 1

If your $f_i$ all satisfy $f_i(0) = 0$, then yes, you can. Divide both sides of the $i$th equation by $x_i$, and bring the $x_i$ term over, leaving only the constant term on the right. Then you have a linear equation in the $x_i$, and assuming that it isn't degenerate, there is a unique solution. However, there will also be additional solutions to the original equation: choose any subset of the $x_i$ to set to $0$. Say that $k$ of them are to be $0$, and the rest non-zero. Set the $k$ values of $x_i$ to $0$ in the original equations, and discard the equations that are now "$0 = 0$". With the remaining equations, you can divide by $x_i$ and proceed as described before.

If any of the $f_i$ does not satisfy $f_i(0) = 0$, then I know of know simple solution to the problem.