Consider a collection of $m$ matrices $A_i$ of size $n\times n$, and a vector $b$ of size $m$. Collecting variables into two vectors $x,y$ of size $n$, I have the bilinear system of equations:
$$\left\{ x^T A_i y = b_i : i = 1,\dots,m \right\}.$$
I would like to consider the domain of the variables to be the complex numbers.
What techniques are available to prove such a system has no solution?
In case it helps, the particular problem I am interested in has $A_i$ and $b$ real valued.
Furthermore, the $A_i$ are such for any pair of values of $j$, $k$ the term $x_j y_k$ will only be used by at most one $A_i$ constraint. (The latter sounds quite constraining, but I couldn't figure out how to use it effectively.
Here is an interesting approach suggested on mathoverflow for solving such systems:
https://mathoverflow.net/questions/308163/solving-system-of-bilinear-equations
The suggestion is to linearize it into a system of equations on the elements of the matrix $Z = x y^T$. Such a matrix (as long as zero is not a solution) must have rank 1. Then we may use convex optimization: the constraints are now linear, and optimize the solution with the objective to minimize $||Z||_{*}$ the nuclear norm which is a proxy for the rank.
I do not understand this method well enough to understand if it is guaranteed to work if there is a solution. In particular, I'd like to invert the argument: if I optimize and get a matrix with rank>1, can I therefore conclude that there is no solution to the original problem?