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2 corrected a few typos

Let $d=\deg S$. If $d\ge 2$ and $S$ is non-degenerate near infinity, in the sense that $|\nabla S(x)|^2 \ge c|x|^{2d-2}$ for $|x|\ge R$ for some positive $R$ and $c$ then the integral converges in a sense close to what you propose. Namely, $$\lim_{r\rightarrow \infty} \int_{\mathbb R^n} \phi(|x|/r) e^{iS(x)} dx$$ exists where $\phi$ is any sufficiently smooth function with $\phi(t)\equiv 1$ for $0\le t\le 1$ and $\phi (t)\equiv 0$ for $t\ge 2$.

To prove this use integration by parts. Fix $r < r'$ and let $$I_{r,r'} = \int_{\mathbb R^n} \phi_{r,r'}(x) e^{i S(x)}dx,$$where $\phi_{r,r'}(x)=\phi(|x|/r)-\phi(|x|/r')$. Assume $r >R$. Multiply and divide by $|\nabla S(x)|^2$, noting that $$|\nabla S(x)|^2e^{i S(x)}= -i\nabla S(x)\cdot \nabla e^{i S(x)}.$$
So we have, after IBP, $$I_{r,r'} = -i \int_{\mathbb R^n} \left [ \nabla \cdot \left ( \frac{\phi_{r,r'}(x)}{|\nabla S(x)|^2} \nabla S(x) \right ) \right ] e^{i S(x)} d x.$$ The integrand is supported in the region $r \le |x| \le 2 r'$, and you can check that it is bounded in magnitude by some constant times $|x|^{-d}$. |x|^{1-d}$. (It is useful to note that second derivatives an$m$-th derivative of$S$are is bounded from above by$|x|^{d-2}$|x|^{d-m}$ since they are polynomials it is a polynomial of degree $d-2$.) If $d$ is large enough this may be enough to control the integral. If not repeat the procedure as many times as necessary to produce a factor that is integrable and you can use to show that $I_{r,r'}$ is small for $r,r'$ sufficiently large. Basically, each time you integrate by parts after multiplying and dividing by $|\nabla S(x)|^2$ you produce a an extra factor of size $|x|^{d-2}$. |x|^{1-d}\$.

All of the above can be extended to integrals with $S$ not necessarily a polynomial, but sufficiently smooth in a neighborhood of infinity and with derivatives that satisfy suitable estimates. The real difficulty with such integrals is not proving that they exist, but estimating their size. Here stationary phase is useful, when applicable, but I don't know of much else.

1

Let $d=\deg S$. If $S$ is non-degenerate near infinity, in the sense that $|\nabla S(x)|^2 \ge c|x|^{2d-2}$ for $|x|\ge R$ for some positive $R$ and $c$ then the integral converges in a sense close to what you propose. Namely, $$\lim_{r\rightarrow \infty} \int_{\mathbb R^n} \phi(|x|/r) e^{iS(x)} dx$$ exists where $\phi$ is any sufficiently smooth function with $\phi(t)\equiv 1$ for $0\le t\le 1$ and $\phi (t)\equiv 0$ for $t\ge 2$.

To prove this use integration by parts. Fix $r < r'$ and let $$I_{r,r'} = \int_{\mathbb R^n} \phi_{r,r'}(x) e^{i S(x)}dx,$$where $\phi_{r,r'}(x)=\phi(|x|/r)-\phi(|x|/r')$. Assume $r >R$. Multiply and divide by $|\nabla S(x)|^2$, noting that $$|\nabla S(x)|^2e^{i S(x)}= -i\nabla S(x)\cdot \nabla e^{i S(x)}.$$
So we have, after IBP, $$I_{r,r'} = -i \int_{\mathbb R^n} \left [ \nabla \cdot \left ( \frac{\phi_{r,r'}(x)}{|\nabla S(x)|^2} \nabla S(x) \right ) \right ] e^{i S(x)} d x.$$ The integrand is supported in the region $r \le |x| \le 2 r'$, and you can check that it is bounded in magnitude by some constant times $|x|^{-d}$. (It is useful to note that second derivatives of $S$ are bounded by $|x|^{d-2}$ since they are polynomials of degree $d-2$.) If $d$ is large enough this may be enough to control the integral. If not repeat the procedure as many times as necessary to produce a factor that is integrable and you can use to show that $I_{r,r'}$ is small for $r,r'$ sufficiently large. Basically, each time you integrate by parts after multiplying and dividing by $|\nabla S(x)|^2$ you produce a an extra factor of size $|x|^{d-2}$.

All of the above can be extended to integrals with $S$ not necessarily a polynomial, but sufficiently smooth in a neighborhood of infinity and with derivatives that satisfy suitable estimates. The real difficulty with such integrals is not proving that they exist, but estimating their size. Here stationary phase is useful, when applicable, but I don't know of much else.
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