As a homogeneous polynomial in indeterminates defines a hypersurface in the projective space of dimension , solving a homogeneous Diophantine equation is the same as finding the rational points of a projective hypersurface.
as A HoMOGenEouS polyNomial in iNdEteRmiNatEs definES a hypERSURface iN tHe pROjEctiVe Space of dimEnsion , soLving A homOGeneoUs dioPHantine EQUaTioN iS the Same as finding the RAtIOnal pOInTs of A pRojecTive hYpErsurFAce.