If $p$ is a positive multivariate polynomial, does $1/p$ have polynomial growth?
Yes. By Stengle's Positivstellensatz, we can find polynomials $f_1$ and $f_2$, such that each of $f_1$ and $f_2$ can be written as sums of squares of polyomials, and $p f_1 = 1+f_2$. (In Wikipedia's language, take $F = \emptyset$ and $W = \mathbb{R}^n$.) Then $1/p = f_1/(1+f_2) \leq f_1$, which is a polynomial.