I have posted this problem in MSE long ago:
<https://math.stackexchange.com/questions/3782868/multi-variable-rational-fraction-integral>. But it hasn't been solved yet so I post it here. Maybe this problem not so easy. I would like to describe it again.


Consider the polynomial function in $\mathbb{R}^n$: $$f(x)=\sum\_{|\alpha|=1}^{m}c\_\alpha x^{\alpha}$$
where $\alpha=(\alpha\_1,\cdots,\alpha\_n)\in \mathbb{N}^n$ is the multi-index with non-negative integers and $|\alpha|=\alpha\_1+\cdots+\alpha\_n$.


**Problem:** For given $p>0$ such that $|\alpha|p<1$ for any $\alpha$, show that in an open bounded domain $D\subset \mathbb{R}^n$
$$\int\_{D}\frac{dx}{|f(x)|^p}<+\infty.$$
**Example to understand:**


(1)
$$\int\_{(-1,1)^2}\frac{dxdy}{|x^3-xy^3+y^2|^{1/5}}$$


(2) For any $0<\varepsilon<1$
$$\int\_{(-1,1)^3}\frac{dxdydz}{|xyz+z^4-y^3x+y^2z|^{(1-\varepsilon)/4}}. $$


**Attempt:** When the case is special I can show this. For example, if $f(x)=x\_1+\sum\limits\_{~~~~~~|\alpha|=1\\ \text{no $x\_1$ appears}}^{m}c\_\alpha x^{\alpha}$, then I can take $f(x)=u\_1$ and $x\_i=u\_i$. The Jacobi is $1$ and we can prove it. But in general it doesn't work.


On the other hand, I have tried the special version of *Resolution of Singularities* (see, e.g. page 147 in Atiyah's <https://onlinelibrary.wiley.com/doi/epdf/10.1002/cpa.3160230202>). But this thoery only makes $f$ become monomial locally with abstract power $\beta\_i$ ($g$ is the blow-down map):
$$f\circ g(u)=c(u)u\_1^{\beta\_1}\cdots u\_n^{\beta\_n}.$$
Since the power $\beta\_i$'s are abstract, the condition $|\alpha|p<1$ doesn't work.


I also tried to apply some theorem in real analysis. For example:



> 
> Suppose $f\geq 0$ and is finite almost everywhere. Let $E\_k=\{x:f(x)>2^k\}$, then $f$ is integrable if and only if $$\sum\_{k=-\infty}^{\infty}2^km(E\_k)<\infty. $$
> One can find this proposition (as an exercise) in Stein and Shakarchi's Real Analysis.
> 
> 
> 


Back to this problem. $E\_k=\{x:\frac{1}{|f(x)|^p}>2^k\}=\{x:|f(x)|<2^{-\frac{k}{p}}\}$. But I'm stuck in estimating the Lebesgue measure of $E\_k$. Is there any way of reference to finish it?


Or you can give some new ways to show this problem. Thank you.



Prove the integral of multi-variable rational fraction is convergent
I'm questioner. Now I have found a reference:


[Volume estimates of sublevel sets of real polynomials](https://arxiv.org/pdf/1711.04544.pdf)


Theorem 4.1 in it can solve this proplem.


This paper has been published at
[Annales Polonici Mathematici](https://www.impan.pl/en/publishing-house/journals-and-series/annales-polonici-mathematici/all/121/2/112545/volume-estimates-of-sublevel-sets-of-real-polynomials)


