TITLE: A system of non-linear equations that is decomposable as a product -- uniqueness of solution?
QUESTION [7 upvotes]: I have a system of non-linear equations 
$ a_1=f_0 g_1$
$a_2=f_1 g_1 + f_0 g_2$
$a_3=f_2 g_1 + f_6 g_2 + f_0 g_3 $
$a_4=f_3 g_1 + f_7 g_2 + f_6 g_3 + f_0 g_4 $
$a_5=f_4 g_1 + f_8 g_2 + f_7 g_3 + f_6 g_4 + f_0 g_5$
$a_6=f_5 g_1 + f_9 g_2 + f_8 g_3 + f_7 g4 + f_6 g_5 + f_0 g_6$
$a_7=f_5 g_3 + f_9 g_4 + f_8 g_5 + f_2 g_6 + f_0 g_7 $
$a_8=f_5 g_5 + f_4 g_6 + f_7 g_7 + f_0 g_8$
$a_9=f_5 g_7 + f_8 g_8 + f_0 g_9$
$a_{10}=f_0 g_{10} + f_5 g_9$
here both $f=(f_0,..,f_5,,..f_9)$ and $g=(g_1,…,g_{10})$ are unknown.
My problem is to figure out when this system has no solution, a unique or multiple solutions.
What is known is that this system of equations allows to decompose itself as
$\left(\begin{array}{c}  a_1\\ a_2 \\ \vdots \\a_9 \\a_{10} \end{array} \right)
 = 
\begin{bmatrix} 
f_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 
f_1 & f_0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\
f_2 & f_6& f_0&  0& 0& 0& 0& 0 &0 & 0 \\
f_3 & f_7 & f_6 & f_0 & 0 & 0 & 0 & 0 & 0&  0 \\
f_4 & f_8 & f_7 & f_6 & f_0&  0& 0& 0& 0& 0 \\ 
f_5 & f_9 & f_8 & f_7 & f_6&  f_0& 0& 0& 0& 0 \\ 
0 & 0 & f_5 & f_9 & f_8 & f_2&  f_0&  0 & 0&  0 \\ 
0 & 0 & 0 & 0 & f_5 & f_4 & f_7 & f_0 & 0 & 0 \\ 
0 & 0 & 0 & 0 & 0 & 0 & f_5& f_8 & f_0 & 0 \\ 
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & f_5 & f_0 
\end{bmatrix} 
\times 
\left( \begin{array}{c} g_1 \\ g_2 \\ \vdots \\ g_9 \\ g_{10} \end{array} \right) $
Then whenever I take only vector of $f$ as unknowns (and so $g$ are the parameters), or vice versa, there is a unique solution for a generic vector $a$, similarly to how it works for linear systems of equations. While when I am letting one extra element of $f$ or of $g$ be "a parameter", i.e. there are more equations than there are unknown $f$ or unknown $g$, there is no solution.
But I have no idea what happens when both $f$ and $g$ are unknown. I have then 10 equations and 20 unknowns, but it is not clear to me that it automatically should lead to a continuum of solutions. (And I cannot solve this system explicitly..) 
If there is a continuum of solutions (for a "generic" $a$), it seems to be an interesting case: a system of $N$ non-linear equations and $M(>N)$ unknowns does not have a solution (for generic vector $a$) whenever the size of the smallest of two sets of unknowns is less than $N$. Or it has a continuum of solutions in the complementary case, but there is no case in-between -- with a unique solution. 
Are there any known results for such system of non-linear equations that are "decomposable"?
I would appreciate any opinions on this!

REPLY [8 votes]: The linear-in-$g$ system is uniquely solvable whenever $f_0\ne0$. therefore the solution set is parametrized by $f$ in the form 
$$g=f_0^{-10}P(f;a),$$
where $P$ is linear in $a$, polynomial in $f$ and homogenenous of degree $9$. Not only it is a continuum, but it has dimension $10$.
When $f_0=0$, there is no solution in general, say for $a_1\ne0$.