  Given a polynomial $f(x_1,x_2,\ldots, x_t)$ in $t$ variables with integer
coefficients and a positive integer $n$, let $\alpha(n)$ be the number of
integers $0\leq a<n$ such that the polynomial congruence $f(x_1, x_2, \ldots,
x_t)\equiv a\ (mod\ n)$ is solvable. We describe a method that allows to
determine the function $\alpha$ associated to polynomials of the form
$c_1x_1^k+c_2x_2^k+\cdots+c_tx_t^k$. Then we apply this method to polynomials
that involve sums and differences of squares, mainly to the polynomials
$x^2+y^2, x^2-y^2$ and $x^2+y^2+z^2$.
