  For a tuple $(\theta_1,..,\theta_M)$ of complex number, buliding on the
approximation techniques in earlier papers of this series, this paper engages
in deducing lower estimates on the transcendence degree of the field generated
by $\theta_1, ..., \theta_M$ over the field of rational numbers from the
approximability of the point $\theta=(1,\theta_1,...,\theta_M)$ in projective
space by hypersurfaces. The first given result is an new proof of an algebraic
independence criterion, that was formerly proved by Laurent and Roy, and
generalizes the Philippon criterion by introducing also evaluations of
derivatives of global sections. The second result is a new kind of algebraic
independence criteria that has a wider range of applicabilty than the first
one.
