Let $\mathcal{P}_d$ denote the space of all real polynomials of degree at most $d$ . It is an old result of Stein and Wainger that

$$\sup_ {P\in\mathcal{P}_d} \bigg\vert p.v.\int_\mathbb{R}{e^{iP(t)}\frac{dt}{t}}\bigg\vert\leq C_d$$

for some constant $C_d$ depending only on $d$ . On the other hand, Carbery, Wainger and Wright claim that the true order of magnitude of the above principal value integral is $\log d$ . We prove that

$$\sup_ {P\in\mathcal{P}_d}\bigg\vert p.v.\int_\mathbb{R}{e^{iP(t)}\frac{dt}{t}}\bigg\vert\sim \log{d}.$$