On minor prime factorizations for multivariate polynomial matrices
Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. In this paper, minor prime factorizations for multivariate polynomial matrices are studied. We give a ne...
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| Published in | Multidimensional systems and signal processing Vol. 30; no. 1; pp. 493 - 502 |
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| Abstract | Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. In this paper, minor prime factorizations for multivariate polynomial matrices are studied. We give a necessary and sufficient condition for the existence of a minor left prime factorization for a multivariate polynomial matrix. This result is a generalization of a theorem in Wang and Kwong (Math Control Signals Syst 17(4):297–311,
2005
). On the basis of this result and a method in Fabiańska and Quadrat (Radon Ser Comp Appl Math 3:23–106,
2007
), we give an algorithm to decide if a multivariate polynomial matrix has minor left prime factorizations and compute one if they exist. |
|---|---|
| AbstractList | Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. In this paper, minor prime factorizations for multivariate polynomial matrices are studied. We give a necessary and sufficient condition for the existence of a minor left prime factorization for a multivariate polynomial matrix. This result is a generalization of a theorem in Wang and Kwong (Math Control Signals Syst 17(4):297–311,
2005
). On the basis of this result and a method in Fabiańska and Quadrat (Radon Ser Comp Appl Math 3:23–106,
2007
), we give an algorithm to decide if a multivariate polynomial matrix has minor left prime factorizations and compute one if they exist. Multivariate polynomial matrix factorizations have been widely investigated during the past years due to the fundamental importance in the areas of multidimensional systems and signal processing. In this paper, minor prime factorizations for multivariate polynomial matrices are studied. We give a necessary and sufficient condition for the existence of a minor left prime factorization for a multivariate polynomial matrix. This result is a generalization of a theorem in Wang and Kwong (Math Control Signals Syst 17(4):297–311, 2005). On the basis of this result and a method in Fabiańska and Quadrat (Radon Ser Comp Appl Math 3:23–106, 2007), we give an algorithm to decide if a multivariate polynomial matrix has minor left prime factorizations and compute one if they exist. |
| Author | Guan, Jiancheng Ouyang, Baiyu Li, Weiqing |
| Author_xml | – sequence: 1 givenname: Jiancheng orcidid: 0000-0002-0418-5378 surname: Guan fullname: Guan, Jiancheng email: jiancheng_guan@aliyun.com organization: College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry Of Education of China), Hunan Normal University – sequence: 2 givenname: Weiqing surname: Li fullname: Li, Weiqing organization: College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry Of Education of China), Hunan Normal University – sequence: 3 givenname: Baiyu surname: Ouyang fullname: Ouyang, Baiyu organization: College of Mathematics and Computer Science, Key Laboratory of High Performance Computing and Stochastic Information Processing (Ministry Of Education of China), Hunan Normal University |
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| Cites_doi | 10.1007/s10440-012-9791-2 10.1109/TCS.1982.1085085 10.1109/PROC.1977.10582 10.1109/TCSII.2005.850516 10.1093/imamci/16.3.275 10.1016/j.jalgebra.2003.10.035 10.1016/j.laa.2004.04.020 10.1109/TCS.1979.1084614 10.1090/gsm/003 10.1016/S0024-3795(01)00370-6 10.1109/TCSI.2007.897720 10.1016/j.laa.2014.09.027 10.1023/A:1008427830183 10.1023/A:1008256224288 10.23919/ECC.2001.7076157 10.1137/S0363012902417127 10.1007/s00498-005-0155-6 |
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| References | WangMOn factor prime factorizations for n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matricesIEEE Transactions on Circuits and Systems I: Regular Papers200754613981405237059610.1109/TCSI.2007.8977201374.15026 EisenbudDCommutative algebra: with a view toward algebraic geometry2013New YorkSpringer0819.13001 FornasiniEValcherMEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matrices with applications to multidimensional signal analysisMultidimensional Systems and Signal Process199729387408146868810.1023/A:10082562242880882.93038 WangMKwongCPOn multivariate polynomial matrix factorization problemsMathematics of Control, Signals, and Systems2005174297311217732010.1007/s00498-005-0155-61098.93010 PommaretJFQuadratAAlgebraic analysis of linear multidimensional control systemsIMA Journal of Mathematical Control and Information199916275297170665810.1093/imamci/16.3.2751158.93319 QuadratAThe fractional representation approach to synthesis problems: An algebraic analysis viewpoint part I: (Weakly) doubly coprime factorizationsSIAM Journal on Control and Optimization2003421266299198274510.1137/S03630129024171271035.93017 LinZXuLFanHOn minor prime factorization for n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matricesIEEE Transactions on Circuits and Systems II: Express Briefs200552956857110.1109/TCSII.2005.850516 LiuJWangMFurther remarks on multivariate polynomial matrix factorizationsLinear Algebra and its Applications2015465204213327467110.1016/j.laa.2014.09.0271310.15021 MatsumuraHReidMCommutative ring theory1989CambridgeCambridge University Press GuiverJPBoseNKPolynomial matrix primitive factorization over arbitrary coefficient field and related resultsIEEE Transactions on Circuits and Systems CAS1982291064965768727910.1109/TCS.1982.10850850504.65020 SrinivasVA generalized Serre problemJournal of Algebra20042782621627207165610.1016/j.jalgebra.2003.10.0351079.13005 BoseNKBuchbergerBGuiverJPApplied multidimensional systems theory2003DordrechtKluwer BrownWCMatrices over commutative rings1993New YorkMarcel Dekker Inc0782.15001 YoulaDCGnaviGNotes on n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-dimensional system theoryIEEE Transactions on Circuits and Systems19792610511152165710.1109/TCS.1979.10846140394.93004 MorfMLevyBCKungSYNew results in 2-D systems theory, Part I: 2-D polynomial matrices, factorization, and coprimenessProceedings of the IEEE197765486187210.1109/PROC.1977.10582 FabiańskaAQuadratAApplications of the Quillen-Suslin theorem to multidimensional systems theoryRadon Series Computational and Applied Mathematics200732310624027091197.13011 AdamsWWLoustaunauPAn introduction to Gröbner bases1994ProvidenceAmerican Mathematical Society10.1090/gsm/0030803.13015 RotmanJJAn introduction to homological algebra2008New YorkSpringer1157.18001 Decker, W., Greuel, G. M., Pfister, G., & Schönemann, H. (2015). Singular 4-0-2—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de. Accessed 20 Dec 2016. Pommaret, J. F. (2001). Solving Bose conjecture on linear multidimensional systems. In Proceedings of the European control conference (pp. 1853–1855). WangMFengDOn Lin–Bose problemLinear Algebra and its Applications2004390279285208365910.1016/j.laa.2004.04.0201056.15013 QuadratAGrade filtration of linear functional systemsActa Applicandae Mathematicae20131272786310114310.1007/s10440-012-9791-21327.16037 LinZNotes on n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matrix factorizationMultidimensional Systems and Signal Process.1999104379393172863510.1023/A:10084278301830939.93017 LinZBoseNKA generalization of Serre’s conjecture and some related issuesLinear Algebra and its Applications2001338125138186111710.1016/S0024-3795(01)00370-61017.13006 A Quadrat (566_CR17) 2003; 42 Z Lin (566_CR10) 2001; 338 M Wang (566_CR22) 2004; 390 A Fabiańska (566_CR6) 2007; 3 JP Guiver (566_CR8) 1982; 29 M Morf (566_CR14) 1977; 65 JF Pommaret (566_CR16) 1999; 16 V Srinivas (566_CR20) 2004; 278 M Wang (566_CR23) 2005; 17 WW Adams (566_CR1) 1994 JJ Rotman (566_CR19) 2008 566_CR4 DC Youla (566_CR24) 1979; 26 WC Brown (566_CR3) 1993 566_CR15 Z Lin (566_CR9) 1999; 10 Z Lin (566_CR11) 2005; 52 NK Bose (566_CR2) 2003 J Liu (566_CR12) 2015; 465 E Fornasini (566_CR7) 1997; 29 M Wang (566_CR21) 2007; 54 D Eisenbud (566_CR5) 2013 A Quadrat (566_CR18) 2013; 127 H Matsumura (566_CR13) 1989 |
| References_xml | – reference: AdamsWWLoustaunauPAn introduction to Gröbner bases1994ProvidenceAmerican Mathematical Society10.1090/gsm/0030803.13015 – reference: FornasiniEValcherMEn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matrices with applications to multidimensional signal analysisMultidimensional Systems and Signal Process199729387408146868810.1023/A:10082562242880882.93038 – reference: LiuJWangMFurther remarks on multivariate polynomial matrix factorizationsLinear Algebra and its Applications2015465204213327467110.1016/j.laa.2014.09.0271310.15021 – reference: EisenbudDCommutative algebra: with a view toward algebraic geometry2013New YorkSpringer0819.13001 – reference: MatsumuraHReidMCommutative ring theory1989CambridgeCambridge University Press – reference: Pommaret, J. F. (2001). Solving Bose conjecture on linear multidimensional systems. In Proceedings of the European control conference (pp. 1853–1855). – reference: PommaretJFQuadratAAlgebraic analysis of linear multidimensional control systemsIMA Journal of Mathematical Control and Information199916275297170665810.1093/imamci/16.3.2751158.93319 – reference: YoulaDCGnaviGNotes on n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-dimensional system theoryIEEE Transactions on Circuits and Systems19792610511152165710.1109/TCS.1979.10846140394.93004 – reference: RotmanJJAn introduction to homological algebra2008New YorkSpringer1157.18001 – reference: BrownWCMatrices over commutative rings1993New YorkMarcel Dekker Inc0782.15001 – reference: Decker, W., Greuel, G. M., Pfister, G., & Schönemann, H. (2015). Singular 4-0-2—A computer algebra system for polynomial computations. http://www.singular.uni-kl.de. Accessed 20 Dec 2016. – reference: FabiańskaAQuadratAApplications of the Quillen-Suslin theorem to multidimensional systems theoryRadon Series Computational and Applied Mathematics200732310624027091197.13011 – reference: WangMKwongCPOn multivariate polynomial matrix factorization problemsMathematics of Control, Signals, and Systems2005174297311217732010.1007/s00498-005-0155-61098.93010 – reference: LinZNotes on n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matrix factorizationMultidimensional Systems and Signal Process.1999104379393172863510.1023/A:10084278301830939.93017 – reference: SrinivasVA generalized Serre problemJournal of Algebra20042782621627207165610.1016/j.jalgebra.2003.10.0351079.13005 – reference: WangMOn factor prime factorizations for n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matricesIEEE Transactions on Circuits and Systems I: Regular Papers200754613981405237059610.1109/TCSI.2007.8977201374.15026 – reference: WangMFengDOn Lin–Bose problemLinear Algebra and its Applications2004390279285208365910.1016/j.laa.2004.04.0201056.15013 – reference: LinZBoseNKA generalization of Serre’s conjecture and some related issuesLinear Algebra and its Applications2001338125138186111710.1016/S0024-3795(01)00370-61017.13006 – reference: LinZXuLFanHOn minor prime factorization for n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-D polynomial matricesIEEE Transactions on Circuits and Systems II: Express Briefs200552956857110.1109/TCSII.2005.850516 – reference: GuiverJPBoseNKPolynomial matrix primitive factorization over arbitrary coefficient field and related resultsIEEE Transactions on Circuits and Systems CAS1982291064965768727910.1109/TCS.1982.10850850504.65020 – reference: QuadratAThe fractional representation approach to synthesis problems: An algebraic analysis viewpoint part I: (Weakly) doubly coprime factorizationsSIAM Journal on Control and Optimization2003421266299198274510.1137/S03630129024171271035.93017 – reference: BoseNKBuchbergerBGuiverJPApplied multidimensional systems theory2003DordrechtKluwer – reference: MorfMLevyBCKungSYNew results in 2-D systems theory, Part I: 2-D polynomial matrices, factorization, and coprimenessProceedings of the IEEE197765486187210.1109/PROC.1977.10582 – reference: QuadratAGrade filtration of linear functional systemsActa Applicandae Mathematicae20131272786310114310.1007/s10440-012-9791-21327.16037 – volume: 127 start-page: 27 year: 2013 ident: 566_CR18 publication-title: Acta Applicandae Mathematicae doi: 10.1007/s10440-012-9791-2 – volume: 29 start-page: 649 issue: 10 year: 1982 ident: 566_CR8 publication-title: IEEE Transactions on Circuits and Systems CAS doi: 10.1109/TCS.1982.1085085 – volume: 65 start-page: 861 issue: 4 year: 1977 ident: 566_CR14 publication-title: Proceedings of the IEEE doi: 10.1109/PROC.1977.10582 – volume: 3 start-page: 23 year: 2007 ident: 566_CR6 publication-title: Radon Series Computational and Applied Mathematics – volume: 52 start-page: 568 issue: 9 year: 2005 ident: 566_CR11 publication-title: IEEE Transactions on Circuits and Systems II: Express Briefs doi: 10.1109/TCSII.2005.850516 – volume: 16 start-page: 275 year: 1999 ident: 566_CR16 publication-title: IMA Journal of Mathematical Control and Information doi: 10.1093/imamci/16.3.275 – volume: 278 start-page: 621 issue: 2 year: 2004 ident: 566_CR20 publication-title: Journal of Algebra doi: 10.1016/j.jalgebra.2003.10.035 – volume-title: Applied multidimensional systems theory year: 2003 ident: 566_CR2 – volume: 390 start-page: 279 year: 2004 ident: 566_CR22 publication-title: Linear Algebra and its Applications doi: 10.1016/j.laa.2004.04.020 – volume: 26 start-page: 105 year: 1979 ident: 566_CR24 publication-title: IEEE Transactions on Circuits and Systems doi: 10.1109/TCS.1979.1084614 – volume-title: An introduction to Gröbner bases year: 1994 ident: 566_CR1 doi: 10.1090/gsm/003 – volume: 338 start-page: 125 year: 2001 ident: 566_CR10 publication-title: Linear Algebra and its Applications doi: 10.1016/S0024-3795(01)00370-6 – volume: 54 start-page: 1398 issue: 6 year: 2007 ident: 566_CR21 publication-title: IEEE Transactions on Circuits and Systems I: Regular Papers doi: 10.1109/TCSI.2007.897720 – volume: 465 start-page: 204 year: 2015 ident: 566_CR12 publication-title: Linear Algebra and its Applications doi: 10.1016/j.laa.2014.09.027 – volume: 10 start-page: 379 issue: 4 year: 1999 ident: 566_CR9 publication-title: Multidimensional Systems and Signal Process. doi: 10.1023/A:1008427830183 – volume: 29 start-page: 387 year: 1997 ident: 566_CR7 publication-title: Multidimensional Systems and Signal Process doi: 10.1023/A:1008256224288 – volume-title: An introduction to homological algebra year: 2008 ident: 566_CR19 – volume-title: Commutative ring theory year: 1989 ident: 566_CR13 – ident: 566_CR15 doi: 10.23919/ECC.2001.7076157 – ident: 566_CR4 – volume-title: Matrices over commutative rings year: 1993 ident: 566_CR3 – volume-title: Commutative algebra: with a view toward algebraic geometry year: 2013 ident: 566_CR5 – volume: 42 start-page: 266 issue: 1 year: 2003 ident: 566_CR17 publication-title: SIAM Journal on Control and Optimization doi: 10.1137/S0363012902417127 – volume: 17 start-page: 297 issue: 4 year: 2005 ident: 566_CR23 publication-title: Mathematics of 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| Title | On minor prime factorizations for multivariate polynomial matrices |
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