Polynomial in several variables.

Question

Let $p:\mathbb{R}^n \to \mathbb{R}$ be a non-constant polynomial. Then what is the smallest positive $t$ (depending on the degree of $p$) for which $$ \int_{\mathbb{R}^n}\dfrac{1}{|p(x)|^t}dx<∞$$?

Answer 1

We assume $p$ is never $0.$

If the degree of $p$ is $d,$ then $1/|p(x)|$ is on the order of $1/|x|^d$ as $|x|\to \infty.$ Thus the integral converges iff

$$\int_{\mathbb R^n} \left (\frac{1}{|x|^d}\right)^t\,dx <\infty.$$

If $n=1$ this happens iff $dt >1.$ If $n=2,$ this happens iff $dt>2,$ as you can see by integrating in polar coordinates. For $n>2,$ the pattern continues. Check out polar coordinates in $\mathbb R^n$ to see this.