Polynomial integrable on bounded sets

**Sogan** · July 22nd 2011, 12:09 PM

Hello,

i have a few questions about a proof. Let P be a polynomial of degree m :
$P:\mathbb{R}^n ->\mathbb{R} , P(x_1 ,...,x_n)=a_m \prod\limits_{j=1}^{m-1}\ (x_1 - q_j (x_2 ,..., x_n))$

Now we want to show that $(\left|P(x)\right|)^{-\sigma}$ with $\sigma <1/m$ is integrable on bounded sets.
The author argues as follows: For any $A < \infty$ , there is a $B < \infty$ , s.t. for $\sigma <1/m$ we get:
$\int\limits_{-A}^A (\left|P(x)\right|)^{-\sigma} dx_1 \leq B$ for $\left|(x_2 ,.., x_n)\right| \leq A$
So first of all i think, because P is a polynomial it has only finite many zeros. Therefore we can think about the integration area without these singularities. But why does the inequality hold?

edit: excuse me, there was something wrong. Now the proposition is correct. I forgot a minus in the exponent, that is $- \sigma$ instead of $\sigma$ .

Please help me with my problem
Regards

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