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!