Let $n_1, n_2 \geq 1$ be known integer constants.

Suppose that we have the following system of $n$ polynomial inequalities for which we know that there exists a feasible solution $(p_1, p_2) \in (0,1)^2$. Furthermore, we assume that each $\epsilon_i$ is much smaller than the corresponding $\mu_i$ ($\epsilon_i$ actually correspond to error terms).

\begin{align*} \mu_1 - \epsilon_1 &\leq n_1 q_1^1 + n_2 q_2^1 \leq \mu_1 + \epsilon_1 \\ \mu_2 - \epsilon_2 &\leq n_1 q_1^2 + n_2 q_2^2 \leq \mu_2 + \epsilon_2 \\ &\vdots \\ \mu_n - \epsilon_n &\leq n_1 q_1^n + n_2 q_2^n \leq \mu_n + \epsilon_n \\ 0 &\leq q_1, q_2 \leq 1\\ \end{align*}

I know that we can use Renegar's Algorithm [1] to solve this system.

**Question**

If the errors $\epsilon_i$ were $0$ I would need just two equations to find the exact solution of the system $(p_1, p_2)$. I would like to know how the set of the solutions of this system changes when I omit some constraints. Assuming that $n$ is huge I have the intuition that dropping many (let's say $\sqrt{n}$) of the constraints should not have a large "impact" on the feasible set since the dimension of the system is very small ($2$). Is there any way that I could measure the "impact" that deleting a constraint has on the area of the feasible set?

[1] : http://www.mathunion.org/ICM/ICM1990.2/Main/icm1990.2.1595.1606.ocr.pdf