Diophantine Approximation on varieties V: Algebraic independence criteria

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For a tuple (θ₁,..,θₘ) 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 θ₁,..., θₘ over the field of rational numbers from the approximability of the point θ=(1,θ₁,...,θₘ) 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.