{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:34:06Z","timestamp":1787333646586,"version":"build-2736575974"},"reference-count":41,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/501100001809","name":"National Natural Science Foundation of China","doi-asserted-by":"publisher","award":["11401364"],"award-info":[{"award-number":["11401364"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["CMMI-1462408"],"award-info":[{"award-number":["CMMI-1462408"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Matrix Anal. Appl."],"published-print":{"date-parts":[[2016,1]]},"abstract":"<jats:p>In this paper we study multivariate polynomial functions in complex variables and their corresponding symmetric tensor representations. The focus is to find conditions under which such complex polynomials always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, characterize the conditions for such complex polynomials to be real valued, and present their corresponding tensor representations. New notions of eigenvalues\/eigenvectors for complex tensors are introduced, extending similar properties from the Hermitian matrices. Moreover, we study a property of the symmetric tensors, namely, the largest eigenvalue (in the absolute value sense) of a real symmetric tensor is equal to its largest singular value; the result is also known as Banach's theorem. We show that a similar result holds for the complex case as well. Finally, we discuss some applications of the new notion of eigenvalues\/eigenvectors for the complex tensors.<\/jats:p>","DOI":"10.1137\/141002256","type":"journal-article","created":{"date-parts":[[2016,3,24]],"date-time":"2016-03-24T12:43:14Z","timestamp":1458823394000},"page":"381-408","source":"Crossref","is-referenced-by-count":21,"title":["Characterizing Real-Valued Multivariate Complex Polynomials and Their Symmetric Tensor Representations"],"prefix":"10.1137","volume":"37","author":[{"given":"Bo","family":"Jiang","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Zhening","family":"Li","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Shuzhong","family":"Zhang","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2016,3,24]]},"reference":[{"key":"atypb1","doi-asserted-by":"crossref","unstructured":"T. Aittomaki and V. Koivunen,\n                      Beampattern optimization by minimization of quartic polynomial\n                      , in Proceedings of the 2009 IEEE\/SP 15th Workshop on Statistical Signal Processing, 2009, pp. 437-440.","DOI":"10.1109\/SSP.2009.5278546"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1109\/TAES.2013.6404093"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1109\/TSP.2013.2273885"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.4064\/sm-7-1-36-44"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1049\/ip-f-1.1983.0003"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1007\/s00014-003-0769-6"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1016\/j.laa.2011.05.040"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/110834524"},{"key":"atypb9","unstructured":"B. Chen, S. He, Z. Li, and S. 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Zhang,\n                      Approximation Methods for Polynomial Optimization: Models, Algorithms, and Applications\n                      , Springer Briefs in Optimization, Springer, New York, 2012.","DOI":"10.1007\/978-1-4614-3984-4"},{"key":"atypb23","first-page":"129","author":"Lim L.-H.","year":"2005","journal-title":"Multi-Sensor Adaptive Processing"},{"key":"atypb24","doi-asserted-by":"publisher","DOI":"10.1109\/MSP.2010.936019"},{"key":"atypb25","doi-asserted-by":"publisher","DOI":"10.1007\/BF00939948"},{"key":"atypb26","doi-asserted-by":"publisher","DOI":"10.1137\/120892891"},{"key":"atypb27","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.07.064"},{"key":"atypb28","doi-asserted-by":"publisher","DOI":"10.1137\/130935112"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsc.2005.05.007"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1016\/j.jmaa.2006.02.071"},{"key":"atypb31","unstructured":"L. Qi,\n                      The Spectral Theory of Tensors (Rough Version)\n                      , Technical Report, http:\/\/arxiv.org\/abs\/1201.3424 arXiv:1201.3424, 2012."},{"key":"atypb32","doi-asserted-by":"crossref","unstructured":"R. Remmert,\n                      Theory of Complex Functions\n                      , Grad. Texts Math. 122, Springer, New York, 1991.","DOI":"10.1007\/978-1-4612-0939-3"},{"key":"atypb33","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-006-0064-6"},{"key":"atypb34","doi-asserted-by":"publisher","DOI":"10.1137\/110832124"},{"key":"atypb35","doi-asserted-by":"publisher","DOI":"10.1137\/120868323"},{"key":"atypb36","unstructured":"L. Sorber, M. Van Barel, and L. 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Qi,\n                      The Quantum Eigenvalue Problem and Z-Eigenvalues of Tensors\n                      , Technical Report, http:\/\/arxiv.org\/abs\/1205.1342 arXiv:1205.1342, 2012."}],"container-title":["SIAM Journal on Matrix Analysis and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/141002256","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T17:15:12Z","timestamp":1787332512000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/141002256"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2016,1]]},"references-count":41,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2016,1]]}},"alternative-id":["10.1137\/141002256"],"URL":"https:\/\/doi.org\/10.1137\/141002256","relation":{},"ISSN":["0895-4798","1095-7162"],"issn-type":[{"value":"0895-4798","type":"print"},{"value":"1095-7162","type":"electronic"}],"subject":[],"published":{"date-parts":[[2016,1]]}}}