Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-sided inequality $-\epsilon\le f_i(x_1,\ldots,x_n) \le \epsilon$. I am interested in how the volume of solutions of the system of inequalities in the same connected component as $0$ behaves asymptotically as $\epsilon\to 0$. A few examples:

- If $f_1 = x_2$ and $f_2 = x_2-x_1$, then $V(\epsilon) \sim \epsilon^2$.
- If $f_1 = x_2$, $f_2=x_2-x_1$ and $f_2 = x_2+x_1$, then still $V(\epsilon) \sim \epsilon^2$.
- If $f_1 = x_2$ and $f_2 = x_2 - x_1^2$, then $V(\epsilon) \sim \epsilon^{3/2}$.
- If $f_1 = x_1 x_2$, $f_2 = x_1^4$, and $f_3 = x_2^4$, then $V(\epsilon)\sim \epsilon \log(1/\epsilon)$.

It seems that the values of $a$ and $b$ such that $0 < \lim_{\epsilon\to0} V(\epsilon) / [\epsilon^a \log(1/\epsilon)^b] < \infty$ should be extractable from the root's multiplicity structure. Is that true? If so, what is the procedure to determine $a$ and $b$?

This seems like a very natural question, and probably there is some literature on it. However, it seems I simply don't know the right words to search for, and I'm not finding anything useful.