On the distribution of fractional parts of polynomials of two variables

In the paper, upper bounds for sums of the form
$$ \underset{(n_1,n_2)\in\Omega}{\sum\sum}\psi(f(n_1,n_2)), $$
where $\psi(x)=x-[x]-\frac12$, $f(x,y)$ is a polynomial, $(n_1,n_2)\in\mathbb Z^2$, and $\Omega$ is a domain in $\mathbb R^2$, are obtained.
One of the upper bounds is of interest, particularly in connection with a lattice point problem considered in Theorem 2.