Consider equations of the form
\begin{equation} n_1 x^i + n_2 y^i = s_i \end{equation}
where $n_1, n_2 \in \mathbb{N}$ are known coefficients and $s_i > 0$ is a real constant.
Fix two indices $i<k$ and consider the following polynomial system
\begin{align*} n_1 x^i + n_2 y^i = s_i\\ n_1 x^k + n_2 y^k = s_k \end{align*} with solutions $(x_1, y_1), \ldots,(x_{ki}, y_{ki}) \in \mathbb{C}^2$. I know that for this system there exists at least one solution $(x*, y*) \in (0,1)^2$.
Now consider the following perturbed system \begin{align*} n_1 x^i + n_2 y^i = \widetilde{s}_i\\ n_1 x^k + n_2 y^k = \widetilde{s}_k \end{align*} where $|\widetilde{s}_i - s_i | < \epsilon_i,\ |\widetilde{s}_k - \widetilde{s}_k| < \epsilon_k$. Let $(\widetilde{x}_1, \widetilde{y}_1), \ldots, (\widetilde{x}_{ki}, \widetilde{y}_{ki}) \in \mathbb{C^2}$ be the solutions of the perturbed system.
Question
I would like to measure the "impact" of the perturbations of the coefficients $s_i, s_k$ on the solutions of the system. References on similar topics are welcome.
First of all it is better to write the solution to the perturbed system this way
\begin{align*} n_1 \widetilde{x} ^i + n_2 \widetilde{y} ^i = \widetilde{s}_i\\ n_1 \widetilde{x}^k + n_2 \widetilde{y} ^k = \widetilde{s}_k \end{align*}
Where $\widetilde{x}=x(1+\delta_x)$ and $ \widetilde{y}=y(1+\delta_y)$.
When the deltas are little we can rewrite the system as :
\begin{align*} n_1 x^i (1+i\delta_x) + n_2 y^i (1+i\delta_y) = {s}_i+\epsilon_i\\ n_1 x^k (1+k\delta_x)+ n_2 y^k (1+k\delta_y)= {s}_k+\epsilon_k \end{align*}
Simplifying out the known factors
\begin{align*} n_1 x^i i\delta_x+ n_2 y^i i\delta_y= \epsilon_i\\ n_1 x^k k\delta_x+ n_2 y^k k\delta_y= \epsilon_k \end{align*}
This is a first order linear system for $(\delta_x,\delta_y)$. You can solve it and study the range of variation of the deltas as $(x, y)$ vary in $(0,1)^2$.