Hello,

i have a few questions about a proof. Let P be a polynomial of degree m :

$\displaystyle P:\mathbb{R}^n ->\mathbb{R} , P(x_1 ,...,x_n)=a_m \prod\limits_{j=1}^{m-1}\ (x_1 - q_j (x_2 ,..., x_n))$

Now we want to show that $\displaystyle (\left|P(x)\right|)^{-\sigma}$ with $\displaystyle \sigma <1/m$ is integrable on bounded sets.

The author argues as follows: For any $\displaystyle A < \infty$, there is a $\displaystyle B < \infty$ , s.t. for $\displaystyle \sigma <1/m$ we get:

$\displaystyle \int\limits_{-A}^A (\left|P(x)\right|)^{-\sigma} dx_1 \leq B$ for $\displaystyle \left|(x_2 ,.., x_n)\right| \leq A$

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$\displaystyle - \sigma$ instead of $\displaystyle \sigma$.

Please help me with my problem

Regards