Let $P(x_1,\cdots,x_n)$ be a real polynomial of degree $\geq 2$. What are the conditions on $P$ so that $$ I_P:=\int_{\mathbb{R}^n} e^{iP(x)} dx $$ exists as an improper Riemann integral ? Already for $n=1$, I dont know the answer, even though my guess is that in this case, it exists for all $P$ of degree $\geq 2$. Is that true and why ?
A related question: at least for $n=1$, do we have a closed formula for $I_P$ when $P$ is a monomial? For example, Mathematica gives $I_{x^3}=-(2\pi)/\Gamma(-1/3)$ and other complicated expressions involving the $\Gamma$ function for higher degrees. What is behind these formulas?