Geodesic
Effective bounds for the number of solutions of certain diophantine equations
Matematičeskie zametki, Tome 8 (1970) no. 3, pp. 361-371

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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. 
@article{MZM_1970_8_3_a8,
     author = {N. I. Fel'dman},
     title = {Effective bounds for the number of solutions of certain diophantine equations},
     journal = {Matemati\v{c}eskie zametki},
     pages = {361--371},
     publisher = {mathdoc},
     volume = {8},
     number = {3},
     year = {1970},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1970_8_3_a8/}
}
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N. I. Fel'dman. Effective bounds for the number of solutions of certain diophantine equations. Matematičeskie zametki, Tome 8 (1970) no. 3, pp. 361-371. http://geodesic.mathdoc.fr/item/MZM_1970_8_3_a8/