Asymptotic expansions for a class of singular integrals emerging in non-linear wave systems

We find asymptotical expansions as $\nu \to 0$ for integrals of the form ${\int }_{{\mathbb{R}}^d}F(x)\mathrm{/}(\omega (x{)}^2+{\nu }^2)dx$, where sufficiently smooth functions $F$ and $\omega$ satisfy natural assumptions for their behaviour at infinity and all critical points of the function $\omega$ from the set $\{\omega (x)=0\}$ are non-degenerate. These asymptotics play a crucial role when analysing stochastic models for non-linear waves systems. Our result generalizes that of [S. Kuksin, Russ. J. Math. Phys.'2017] where a similar asymptotics was found in a particular case when $\omega$ is a non-degenerate quadratic form of the signature $(d\mathrm{/}2,d\mathrm{/}2)$ with even $d$.