Abstract
We give a local parametric description of all complex hypersurfaces in $${\mathbb{C}^{n+1}}$$ and in complex projective space $${\mathbb{CP}^{n+1}}$$ with constant index of relative nullity, together with applications. This is a complex analogue to the parametrization for real hypersurfaces in Euclidean space known as the Gauss parametrization.
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