# Oscillatory integrals of algebraic functions

Consider an algebraic function $\phi$ on $R^{d}$.

By this I mean that there exists a polynomial $P$ with coefficients in $R[x_1,...,x_d]$ (coefficients are polynomials!) such that $P(\phi) = 0$

Let $r$ be a real number >0, I would like to know whether $1/r^{d}$ times $\int_{[-r,r]^d} e^{i \phi}$ tends to zero when $r$ tends to infinity.

Of course it would be enough to show that this integral is bounded by $r^{d-\epsilon}$...

I am not at all familiar with the subject, but from the literature I looked at, I understand that this is the case when $\phi$ itself is a polynomial. I was wondering whether this holds true for any algebraic function.

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