We consider an overdetermined set of equations, consisting of two equations for one complex variable $x$. I want to show that there are no roots for $x$ in the complex unit disc but without the origin, i.e. for $0 < |x| < 1$.
Consider the example system \begin{align} 0 &= F(x)^2 + A(x), \\ 0 &= F(x) - B(x), \end{align} where $A(x)$, $B(x)$ and $F(x)$ are all polynomials in $x$. The exact expressions for these three variables are unknown. However, we do know that $A(x)$ is at least of the order $x^2$ and $B(x)$ is at least of the order $x$. There is a possibility that $F(x) := 0$ and not a function of $x$ at all.
My proposed technique is taking the second equation and stating that $F(x) = B(x)$ and substituting this in the first equations to obtain \begin{align} 0 = B(x)^2 + A(x). \end{align}
If I can now show that the above equation has no roots in the complex unit disc, I am done. At least, that is what I thought. I can actually prove using Rouche's theorem that the above equation has a number of roots in the complex unit disc. However, none of these roots $x$ make the first two equations zero.
Clearly, there is an error in my reasoning. Could you help to point this out to me? My guess is that I am not allowed to substitute one equation to the other, but I do not know why.