The volume around a singular isolated root when equalities are loosened

Question

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:

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

Answer 1

This question seems to be closely related to the computation of asymptotics of oscillatory integrals, which has been studied extensively, in e.g. Volume 2 of the treatise "Singularities of Differentiable Maps" by Arnol'd, Gusesin-Zade and Varchenko. For a shorter exposition with detailed examples, Liviu Nicolaescsu previously gave an answer on MO linking to the senior thesis of Zach Lamberty.

The connection is roughly as follows, in Lamberty's notation. The oscillatory integrals are typically of the form $$I_\xi(a)=\int_{\mathbb{R}^n}e^{i\phi(x)\xi}a(x)dx,$$ and we are interested in its asymptotics as $\xi\rightarrow\infty$.

Recall the idea of the stationary phase approximation - the primary contribution to this integral comes from the critical points of the phase function $\phi(x)$. Thus in order to compute the asymptotics, one is interested in the behavior of $a$ and $\phi$ on the set $|\phi'(x)|<\epsilon$. It turns out that the resulting asymptotic expansions take forms similar to the ones that you gave: $$f(t)\sim\sum_{a,k}C_{a,k}t^a(\log|t|)^k.$$

Now you are interested in a case where you don't have any oscillatory integral so you can skip directly to looking at these sublevel sets of the phase. However, you have to deal with something like multiple phase functions. I found a paper which appears to cover this case, "On the volume and number of some semialgebraic sets" by Ha Huy Vui and Tran Gia Loc (arXiv link). They also give citations to other literature that might be helpful.