Detraz, Jacqueline; Trepreau, Jean Marie Une charactérisation des quadriques hermitiennes dans \(\mathbb{C}^ n\). (A characterization of Hermite quadrics in \(\mathbb{C}^ n\)). (French) Zbl 0747.35007 J. Anal. Math. 55, 51-58 (1990). 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. Reviewer: J.Siciak (Kraków) Cited in 8 Documents MSC: 35J25 Boundary value problems for second-order elliptic equations 32V40 Real submanifolds in complex manifolds 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.) PDFBibTeX XMLCite \textit{J. Detraz} and \textit{J. M. Trepreau}, J. Anal. Math. 55, 51--58 (1990; Zbl 0747.35007) Full Text: DOI References: [1] Avanissian, V.; Traore, A., Sur les fonctions polyanalytiques de plusieurs variables, C.R., Acad. Sci. Paris, 286, 743-746 (1978) · Zbl 0388.32001 [2] Baouendi, S.; Chang, C.; Trèves, F., Microlocal hypo-analyticity and extensions of C.R. functions, J. Differ. Geom, 18, 331-391 (1983) · Zbl 0575.32019 [3] Bruna, J.; Cufi, Y.; Verdera, J., Cauchy kernels in strictly pseudoconvex domains and an application to a Mergelyan type approximation problem, Math. Z., 189, 41-53 (1985) · Zbl 0538.32003 [4] Detraz, J., Problème de Dirichlet pour le système ∂^2f/∂z_j∂z_k = 0, Ark. Mat., 26, 173-184 (1988) · Zbl 0703.35050 [5] H. Shapiro,Unbounded quadrature domains, inComplex Analysis I, Springer Lecture Notes in Mathematics No. 1275, pp. 287-331. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.