On the distribution of fractional parts of polynomials

In the paper upper bounds for sums of the form
$$ \sum_{0<n\le N}\psi\big(f(n)\big), $$
where $f(x)$ is a polynomial and $\psi(x)=x-[x]-1/2$, are obtained.
The cases
$$ f(x)=\frac1\alpha x^2+\beta x+\gamma $$
and
$$ f(x)=\frac1\alpha x^3+\beta x^2+\gamma x+\delta $$
are considered, where $\alpha$ is a large positive number.
Weyl's method and V. N. Popov's reasoning (Mat. Zametki, 18 (1975), 699–704) are used.