Une charactérisation des quadriques hermitiennes dans \(\mathbb{C}^ n\). (A characterization of Hermite quadrics in \(\mathbb{C}^ n\)) (Q1174675)

Main result: Let \(S:=\{r(z)=0\}\) be a connected hypersurface in \(\mathbb{C}^ n\) of class \({\mathcal C}^ 8\). The following conditions are equivalent (i) If \(r(z)=0\) and \(\sum^ n_{j=1}(\partial r/\partial z_ j)(z)w_ j=0\) then \(\sum^ n_{j,k=1}(\partial^ 2r/\partial z_ j\partial z_ k)(z)w_ jw_ k=0\); (ii) Either \(S\) is contained in a Hermite quadric, or \(S\) can be foliated by complex hyperplanes.