Consider $f_1(t,x_1,\ldots,x_n),\ldots, f_n(t, x_1,\ldots, x_n)$ for complex analytic functions $f_i$ around $\vec{a}=(t', a_1, \ldots, a_n)$ and $\vec{b}=(t^*, b_1,\ldots, b_n)$, such that $f_i(\vec{a})=0$ and $f_i(\vec{b})=0$, and the Jacobian of this system is non-zero at $\vec{a}$ and $\vec{b}$. If I'm missing anything from the analytic implicit function theorem, then assume that stuff too. Anyway, by the IFT, there exists functions $(g_1(t),\ldots, g_n(t))$ analytic around some $\epsilon_1$-neighborhood $D_{\epsilon_1}$ of $\vec{a}$, and functions $(h_1(t),\ldots, h_n(t))$ that are analytic in some $\epsilon_2$-neighborhood $D_{\epsilon_2}$ of $\vec{b}$, such that $f_i(t, g_1(t), \ldots, g_n(t))=0$ for $t\in D_{\epsilon_1}$ and $f_i(t, h_1(t),\ldots, h_n(t))=0$ for $t\in D_{\epsilon_2}$.
Now, assume that $D_{\epsilon_1}\cap D_{\epsilon_2}\neq \emptyset$. When, if ever, is it the case that $g_i(t)=h_i(t)$ on $D_{\epsilon_1}\cap D_{\epsilon_2}$? I would assume this is true when $\vec{a}$ and $\vec{b}$ are "close", i.e., on the same sheet in the covering space, but I don't know how to formalize this notion, nor prove it. Any hints/answers are very much appreciated.
Edit: something something covering map, locally injective?