Effective bounds for the number of solutions of certain diophantine equations

It is proved that the number of solutions of the diophantine equation
$$ \mathrm{Norm}\,(z_1\omega_1+\dots+z_m\omega_m)=f(z_1,\dots,z_m), $$
is finite, where $\omega_1,\dots,\omega_m$ are algebraic numbers of a special type, the left side of the equation is the norm with respect to a quadratic field, and $f$ is a low-degree polynomial.