  We investigate the problem of holomorphic algebraizibility for real
hypersurfaces in complex space. We introduce a new invariant of a
(real-analytic) Levi-nondegenerate hypersurface called {\em the jet
transcendence degree}. Using this invariant, we solve in the negative the
Conjecture of Huang, Ji and Yau on the algabraizability of real hypersurfaces
with algebraic syzygies.
