{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,28]],"date-time":"2026-03-28T01:25:43Z","timestamp":1774661143886,"version":"3.50.1"},"reference-count":33,"publisher":"American Mathematical Society (AMS)","issue":"288","license":[{"start":{"date-parts":[[2014,11,7]],"date-time":"2014-11-07T00:00:00Z","timestamp":1415318400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>A mixed trigonometric polynomial system, which rather frequently occurs in applications, is a polynomial system where every monomial is a mixture of some variables and sine and cosine functions applied to the other variables. Polynomial systems transformed from the mixed trigonometric polynomial systems have a special structure. Based on this structure, a hybrid polynomial system solving method, which is more efficient than random product homotopy method and polyhedral homotopy method in solving this class of systems, has been presented. Furthermore, the transformed polynomial system has an inherent partially symmetric structure, which cannot be adequately exploited to reduce the computation by the existing methods for solving polynomial systems. In this paper, a symmetric homotopy is constructed and, combining homotopy methods, decomposition, and elimination techniques, an efficient symbolic-numerical method for solving this class of polynomial systems is presented. Preservation of the symmetric structure assures us that only part of the homotopy paths have to be traced, and more important, the computation work can be reduced due to the existence of the inconsistent subsystems, which need not to be solved at all. Exploiting the new hybrid method, some problems from the literature and a challenging practical problem, which cannot be solved by the existing methods, are resolved. Numerical results show that our method has an advantage over the polyhedral homotopy method, hybrid method and regeneration method, which are considered as the state-of-art numerical methods for solving highly deficient polynomial systems of high dimension.<\/p>","DOI":"10.1090\/s0025-5718-2013-02763-9","type":"journal-article","created":{"date-parts":[[2013,11,7]],"date-time":"2013-11-07T15:22:40Z","timestamp":1383837760000},"page":"1847-1868","source":"Crossref","is-referenced-by-count":6,"title":["A symmetric homotopy and hybrid polynomial system solving method for mixed trigonometric polynomial systems"],"prefix":"10.1090","volume":"83","author":[{"given":"Bo","family":"Dong","sequence":"first","affiliation":[]},{"given":"Bo","family":"Yu","sequence":"additional","affiliation":[]},{"given":"Yan","family":"Yu","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2013,11,7]]},"reference":[{"key":"1","unstructured":"Daniel J. Bates, Jonathan D. Hauenstein, Andrew J. Sommese, and Charles W. Wampler, Bertini: Software for numerical algebraic geometry, http:\/\/www.nd.edu\/\u223csommese\/bertini\/."},{"key":"2","series-title":"Graduate Texts in Mathematics","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4757-6911-1","volume-title":"Using algebraic geometry","volume":"185","author":"Cox, David","year":"1998","ISBN":"https:\/\/id.crossref.org\/isbn\/0387984879"},{"key":"3","series-title":"Graduate Texts in Mathematics, No. 52","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4757-3849-0","volume-title":"Algebraic geometry","author":"Hartshorne, Robin","year":"1977","ISBN":"https:\/\/id.crossref.org\/isbn\/0387902449"},{"issue":"3-4","key":"4","doi-asserted-by":"publisher","first-page":"323","DOI":"10.1007\/BF02307383","article-title":"Safe starting regions by fixed points and tightening","volume":"53","author":"Hong, H.","year":"1994","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"issue":"212","key":"5","doi-asserted-by":"publisher","first-page":"1541","DOI":"10.2307\/2153370","article-title":"A polyhedral method for solving sparse polynomial systems","volume":"64","author":"Huber, Birkett","year":"1995","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"273","key":"6","doi-asserted-by":"publisher","first-page":"345","DOI":"10.1090\/S0025-5718-2010-02399-3","article-title":"Regeneration homotopies for solving systems of polynomials","volume":"80","author":"Hauenstein, Jonathan D.","year":"2011","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"2-3","key":"7","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1007\/s00607-008-0015-6","article-title":"HOM4PS-2.0: a software package for solving polynomial systems by the polyhedral homotopy continuation method","volume":"83","author":"Lee, T. L.","year":"2008","journal-title":"Computing","ISSN":"https:\/\/id.crossref.org\/issn\/0010-485X","issn-type":"print"},{"key":"8","first-page":"209","article-title":"Numerical solution of polynomial systems by homotopy continuation methods","author":"Li, T. Y.","year":"2003"},{"issue":"5","key":"9","doi-asserted-by":"publisher","first-page":"1241","DOI":"10.1137\/0726069","article-title":"The cheater\u2019s homotopy: an efficient procedure for solving systems of polynomial equations","volume":"26","author":"Li, T. Y.","year":"1989","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"10","doi-asserted-by":"publisher","first-page":"481","DOI":"10.1007\/BF01400351","article-title":"The random product homotopy and deficient polynomial systems","volume":"51","author":"Li, T.-Y.","year":"1987","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"194","key":"11","doi-asserted-by":"publisher","first-page":"693","DOI":"10.2307\/2008402","article-title":"Solving deficient polynomial systems with homotopies which keep the subschemes at infinity invariant","volume":"56","author":"Li, T. Y.","year":"1991","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"12","first-page":"433","article-title":"On Chow, Mallet-Paret and Yorke homotopy for solving system of polynomials","volume":"11","author":"Li, Tien-Yien","year":"1983","journal-title":"Bull. Inst. Math. Acad. Sinica","ISSN":"https:\/\/id.crossref.org\/issn\/0304-9825","issn-type":"print"},{"issue":"3","key":"13","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/BF03023953","article-title":"Solving polynomial systems","volume":"9","author":"Li, Tien-Yien","year":"1987","journal-title":"Math. Intelligencer","ISSN":"https:\/\/id.crossref.org\/issn\/0343-6993","issn-type":"print"},{"issue":"3","key":"14","doi-asserted-by":"publisher","first-page":"251","DOI":"10.11650\/twjm\/1500407124","article-title":"Solving polynomial systems by polyhedral homotopies","volume":"3","author":"Li, Tien-Yien","year":"1999","journal-title":"Taiwanese J. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/1027-5487","issn-type":"print"},{"issue":"3","key":"15","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1007\/BF03167887","article-title":"A simple homotopy for solving deficient polynomial systems","volume":"6","author":"Li, Tien-Yien","year":"1989","journal-title":"Japan J. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0910-2043","issn-type":"print"},{"issue":"2","key":"16","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1137\/0724032","article-title":"Numerical solution of a class of deficient polynomial systems","volume":"24","author":"Li, Tien-Yien","year":"1987","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"17","first-page":"277","article-title":"Symmetric homotopies for solving systems of polynomial equations","volume":"39","author":"Merav\u00fd, Pavol","year":"1989","journal-title":"Math. Slovaca","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5173","issn-type":"print"},{"issue":"2","key":"18","doi-asserted-by":"publisher","first-page":"115","DOI":"10.1016\/0096-3003(87)90064-6","article-title":"Computing all solutions to polynomial systems using homotopy continuation","volume":"24","author":"Morgan, Alexander","year":"1987","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"2","key":"19","doi-asserted-by":"publisher","first-page":"101","DOI":"10.1016\/0096-3003(87)90063-4","article-title":"A homotopy for solving general polynomial systems that respects \ud835\udc5a-homogeneous structures","volume":"24","author":"Morgan, Alexander","year":"1987","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"1","key":"20","doi-asserted-by":"publisher","first-page":"87","DOI":"10.1016\/0096-3003(86)90030-5","article-title":"A homotopy for solving polynomial systems","volume":"18","author":"Morgan, Alexander P.","year":"1986","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"2","key":"21","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/0096-3003(89)90099-4","article-title":"Coefficient-parameter polynomial continuation","volume":"29","author":"Morgan, Alexander P.","year":"1989","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"2","key":"22","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1016\/0096-3003(89)90099-4","article-title":"Coefficient-parameter polynomial continuation","volume":"29","author":"Morgan, Alexander P.","year":"1989","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"2","key":"23","doi-asserted-by":"publisher","first-page":"797","DOI":"10.1137\/S0036142995281504","article-title":"Solving polynomial systems using a branch and prune approach","volume":"34","author":"Van Hentenryck, Pascal","year":"1997","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"24","doi-asserted-by":"crossref","unstructured":"Jan Verschelde, Algorithm 795: Phcpack: A general-purpose solver for polynomial systems by homotopy continuation, 25 (1999), no. 2, 251\u2013276.","DOI":"10.1145\/317275.317286"},{"issue":"3","key":"25","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1016\/0096-3003(91)90059-V","article-title":"A new start system for solving deficient polynomial systems using continuation","volume":"44","author":"Verschelde, Jan","year":"1991","journal-title":"Appl. Math. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0096-3003","issn-type":"print"},{"issue":"1-3","key":"26","doi-asserted-by":"publisher","first-page":"575","DOI":"10.1016\/0377-0427(94)90329-8","article-title":"Symmetric homotopy construction","volume":"50","author":"Verschelde, Jan","year":"1994","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"issue":"1","key":"27","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1006\/aama.1995.1005","article-title":"Symmetric Newton polytopes for solving sparse polynomial systems","volume":"16","author":"Verschelde, Jan","year":"1995","journal-title":"Adv. in Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0196-8858","issn-type":"print"},{"issue":"2","key":"28","doi-asserted-by":"publisher","first-page":"583","DOI":"10.1137\/0730028","article-title":"The \ud835\udc3a\ud835\udc35\ud835\udc44-algorithm for constructing start systems of homotopies for polynomial systems","volume":"30","author":"Verschelde, Jan","year":"1993","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"3","key":"29","doi-asserted-by":"publisher","first-page":"915","DOI":"10.1137\/0731049","article-title":"Homotopies exploiting Newton polytopes for solving sparse polynomial systems","volume":"31","author":"Verschelde, Jan","year":"1994","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"169","key":"30","doi-asserted-by":"publisher","first-page":"125","DOI":"10.2307\/2007797","article-title":"Finding all solutions to a system of polynomial equations","volume":"44","author":"Wright, Alden H.","year":"1985","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3","key":"31","doi-asserted-by":"publisher","first-page":"1503","DOI":"10.1137\/070681740","article-title":"A hybrid polynomial system solving method for mixed trigonometric polynomial systems","volume":"46","author":"Yu, Bo","year":"2008","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"key":"32","first-page":"36","article-title":"The random product homotopy for solving polynomial systems in \ud835\udc36^{\ud835\udc58\u2081}\u00d7\ud835\udc36^{\ud835\udc58\u2082}\u00d7\u22ef\u00d7\ud835\udc36^{\ud835\udc58_{\ud835\udc5a}}","author":"Yu, Bo","year":"1993"},{"issue":"181","key":"33","doi-asserted-by":"publisher","first-page":"167","DOI":"10.2307\/2007920","article-title":"A simple homotopy method for determining all isolated solutions to polynomial systems","volume":"50","author":"Zulehner, Walter","year":"1988","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2013-02763-9\/S0025-5718-2013-02763-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"},{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2013-02763-9\/S0025-5718-2013-02763-9.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,30]],"date-time":"2021-07-30T05:37:16Z","timestamp":1627623436000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2014-83-288\/S0025-5718-2013-02763-9\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,7]]},"references-count":33,"journal-issue":{"issue":"288","published-print":{"date-parts":[[2014,7]]}},"alternative-id":["S0025-5718-2013-02763-9"],"URL":"https:\/\/doi.org\/10.1090\/s0025-5718-2013-02763-9","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["0025-5718","1088-6842"],"issn-type":[{"value":"0025-5718","type":"print"},{"value":"1088-6842","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,7]]}}}