Consider the polynomial function in $R^n$: $$f(x)=\sum_{|\alpha|=1}^{m}c_\alpha x^{\alpha}$$ where $\alpha=(\alpha_1,\cdots,\alpha_n)\in R^n$ 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 a bounded domain $D$ $$\int_{D}\frac{dx}{|f(x)|^p}<+\infty$$.

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 the difficulty is that the zero of $f(x)$ not only in the origin. I'm stuck here.



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