D’Angelo, John P.; Putinar, Mihai Hermitian complexity of real polynomial ideals. (English) Zbl 1273.14120 Int. J. Math. 23, No. 6, 1250026, 14 p. (2012). Summary: We define the Hermitian complexity of a real polynomial ideal and of a real algebraic subset of \(\mathbb C^{n}\). This concept is aimed at determining precise necessary conditions for a Hermitian symmetric polynomial to agree with a Hermitian squared norm on an algebraic set. The latter topic has been a central theme in modern polynomial optimization and in complex geometry, specifically related to the holomorphic embedding of pseudoconvex domain into balls, or the classification of proper holomorphic maps between balls. Cited in 2 Documents MSC: 14P05 Real algebraic sets 15B57 Hermitian, skew-Hermitian, and related matrices 32V15 CR manifolds as boundaries of domains Keywords:Hilbert’s 17th problem; Hermitian forms; sums of squares; Hermitian length; algebraic sets; positivity conditions; CR complexity PDFBibTeX XMLCite \textit{J. P. D'Angelo} and \textit{M. Putinar}, Int. J. Math. 23, No. 6, 1250026, 14 p. (2012; Zbl 1273.14120) Full Text: DOI References: [1] DOI: 10.1090/gsm/044 · doi:10.1090/gsm/044 [2] DOI: 10.2307/1969817 · Zbl 0051.13103 · doi:10.2307/1969817 [3] DOI: 10.4310/MRL.1996.v3.n2.a2 · Zbl 0858.32010 · doi:10.4310/MRL.1996.v3.n2.a2 [4] D’Angelo J. P., Several Complex Variables and the Geometry of Real Hypersurfaces (1991) [5] DOI: 10.5948/UPO9780883859704 · doi:10.5948/UPO9780883859704 [6] DOI: 10.4134/JKMS.2003.40.3.341 · Zbl 1044.32010 · doi:10.4134/JKMS.2003.40.3.341 [7] DOI: 10.1142/S0129167X05002990 · Zbl 1079.32013 · doi:10.1142/S0129167X05002990 [8] DOI: 10.1016/j.aim.2010.12.013 · Zbl 1218.32001 · doi:10.1016/j.aim.2010.12.013 [9] DOI: 10.1007/s12220-010-9160-1 · Zbl 1228.32037 · doi:10.1007/s12220-010-9160-1 [10] DOI: 10.1090/S0002-9939-2011-10841-4 · Zbl 1309.12001 · doi:10.1090/S0002-9939-2011-10841-4 [11] D’Angelo J. P., Asian J. Math. 7 pp 1– [12] DOI: 10.2307/3597203 · Zbl 1033.12001 · doi:10.2307/3597203 [13] J. W. Helton and M. Putinar, Operator Theory, Structured Matrices, and Dilations (2007) pp. 101–176. [14] DOI: 10.1142/S0129167X90000071 · Zbl 0703.32009 · doi:10.1142/S0129167X90000071 [15] DOI: 10.2307/2374211 · Zbl 0499.32014 · doi:10.2307/2374211 [16] DOI: 10.1007/BF01215140 · Zbl 0584.32048 · doi:10.1007/BF01215140 [17] DOI: 10.1007/BF01425382 · Zbl 0222.10022 · doi:10.1007/BF01425382 [18] DOI: 10.4310/MRL.2010.v17.n6.a4 · Zbl 1230.14086 · doi:10.4310/MRL.2010.v17.n6.a4 [19] DOI: 10.1007/BF01389773 · Zbl 0198.35205 · doi:10.1007/BF01389773 [20] DOI: 10.1007/978-0-387-09686-5_8 · doi:10.1007/978-0-387-09686-5_8 [21] DOI: 10.1112/S0024610706022708 · Zbl 1168.32018 · doi:10.1112/S0024610706022708 [22] DOI: 10.1353/ajm.2008.0012 · Zbl 1146.32008 · doi:10.1353/ajm.2008.0012 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.