Say I have a square family of systems $F_p=(f_1,\ldots,f_N)=0$ where each $f_i\in \mathbb{C}[p][x_1,\ldots,x_n]$, for parameter $p$ and variables $x_i$. I also have an $(a, \vec{b})$ pair such that $F_a(\vec{b})=0$.
I know that for (almost) any such $a\in \mathbb{C}$, I can create locally continuous functions $x_i(p)\in \mathbb{C}[p]$ such that $x_i(a)=b_i$. Each such $x_i(p)$ is supposed to (locally) tell the behavior of the $x_i$-th coordinate of $F_p=0$, and so $(x_1(p), \ldots, x_n(p))$ describes the behavior of all of coordinates of $F_p=0$ on some small neighborhood of $a$ .
I'm interested in approximating these $x_i(p)$ so that I have (hopefully slightly larger) regions around $a$ on which I closely approximate the behavior of $F_p$. I can, say, sample points to get such approximations. My question is: if I take some other $\alpha \in \mathbb{C}$ and look at $k=(x_1(\alpha), \ldots, x_n(\alpha))$, how can I tell whether or not ${\alpha}$ is close enough to $a$ to lie in this "good" region? I know I can look at $||F_\alpha(k)||$, or maybe do some Smale $\alpha$-theory, but is there a different way that focuses more on the properties of the $x_i(p)$'s? Perhaps showing that $\alpha$ lies in the radius of convergence of the $x_i(p)$'s (I feel like this requires defining a new notion of "radius of convergence")?
Anything/everything helps :)