# The volume around a singular isolated root when equalities are loosened

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.

• 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 – j.c. Apr 13 '16 at 18:22
• 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. – Yoav Kallus Apr 13 '16 at 18:48

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: $$f(t)\sim\sum_{a,k}C_{a,k}t^a(\log|t|)^k.$$
• 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 $g(y) =\exp [ i (y/t)^2 ]$. – Yoav Kallus Apr 13 '16 at 21:25