In a sense this is a follow up question to The mathematical theory of Feynman integrals although by all rights it should precede that question.

Let $S$ be a polynomial with real coefficients in $n$ variables. Is there a criterion which would say when the integral $$\int_{\mathbf{R}^n}e^{iS(x)}dx$$ converges? Here $dx$ is the Lebesgue measure on $\mathbf{R}^n$ and the integral is understood as the limit of the integrals over the balls of radius $r$ centered at the origin (with respect to the standard metric) as $r\to\infty$.

Some obvious remarks:

If $\deg S=2$, the integral converges if and only if the quadratic part of $S$ is nondegenerate.

If $n=1$, the integral converges if and only if $\deg S>1$.

The answer to the above question is probably classical (but it is unknown to me).

upd: Conjecture (inspired by Jeff's answer below). A sufficient condition for the integral to converge is as follows: let $S_i$ be the degree $i$ part of $S$ and let $V_i,i=1,\ldots,d=\deg S$ be the subvariety of the real projective space $\mathbf{P}^{n-1}(\mathbf{R})$ given by $S_i=0$. The integral converges if $V_2\cap\cdots\cap V_d={\emptyset}$. Here is how one can try to prove this: the above condition is equivalent to saying that the integral along any line converges, so one can try to first integrate along all half-lines emanating from the origin, get a continuous function on the sphere (hopefully) and then integrate it along the sphere. As remark 1. above shows, this condition may be sufficient but it is not necessary.