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.