I have a system of $n$ quadratic equations with $n$ unknowns. It can be written as 


$diag(x)Ax=1$


$x$ is an $n$-vector, $A$ is $n\times n$, real, symmetric and positive definite, the diagonal elements of $A$ are strictly positive, other elements of $A$ are arbitrary, $1$ is a vector of ones and $diag(x)$ is just s shorthand for taking a vector and putting it on the diagonal of a matrix, the rest of the entries are all zeros.


I am interested in the following question:


1. How many strictly positive solutions exist (in general)?


Any help (examples etc). is greatly appreciated!



Positive solutions to simultaneous real quadratic equations
Your system says $(Ax)\_i = x\_i^{-1}$. Suppose you had two distinct, positive solutions $x, y$. Then


$$ \eqalign{(x-y)^T A (x-y) &= \sum\_{i} (x-y)\_i (A (x-y))\_i\cr & = \sum\_i (x\_i-y\_i)(x\_i^{-1} - y\_i^{-1})\cr
&= - \sum\_{i} \dfrac{(x\_i - y\_i)^2}{x\_i y\_i} < 0} $$
which contradicts the assumption that $A$ is positive definite. So there can only be at most one positive solution.


