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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.

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I'm questioner. Now I have found a reference:

Volume estimates of sublevel sets of real polynomials

Theorem 4.1 in it can solve this proplem.

This paper has been published at Annales Polonici Mathematici

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