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

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.

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  • $\begingroup$ I attended a talk last year by Miranda Holmes-Cerfon where she discussed thinking about this problem. I believe the reference she said that she and her collaborators were trying to digest was this one by Chambert-Loir and Tschinkel: worldscientific.com/doi/abs/10.1142/S1793744210000223 $\endgroup$ – j.c. Apr 13 '16 at 18:22
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    $\begingroup$ Thanks @j.c. I am in fact a collaborator of Miranda's, and I perhaps should have wrote so in the question. I suggested to post this question since I refused to believe that the answer was that obscure and I had faith that the MO community would be able to point us in the right direction. $\endgroup$ – Yoav Kallus Apr 13 '16 at 18:48
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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.

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    $\begingroup$ Thanks. Those look like good places to start. The connection to the oscillatory integrals might be even closer than you point out. The way I phrased the question, the integral is $\int_{\mathbb{R}^n} \prod_{i=0}^m g(f_i(x)) dx$, where $g(y) = 1$ for $-t<y<t$ and $0$ otherwise. But if $g(y) = \exp[ - (y/t)^2 ]$, I can show that the answer is the same. I imagine that the answer is still the same if $g(y) =\exp [ i (y/t)^2 ]$. $\endgroup$ – Yoav Kallus Apr 13 '16 at 21:25
  • $\begingroup$ I recently saw a talk by Caroline Uhler about applications of algebraic geometry to statistics where she discussed computing volumes of thickened hypersurfaces. The relevant paper is here: arxiv.org/abs/1209.0285 and they call the pair (a, b) (in the notation of your question) the "real log canonical threshold". You might try asking her too. $\endgroup$ – j.c. Aug 11 '17 at 18:32
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    $\begingroup$ Thanks. Since I posted this question, and thanks in part to your answer, I have already found the connection to the "real log canonical threshold." I found a very informative paper (arxiv.org/abs/1003.5338) by one of the authors of the paper you linked. I'm sorry I forgot to mention this here. $\endgroup$ – Yoav Kallus Aug 11 '17 at 18:38

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