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ZHENG Xiaopeng^{1,2}, LU Dong^{3}, WANG Dingkang^{1,2}, XIAO Fanghui^{4}
ZHENG Xiaopeng, LU Dong, WANG Dingkang, XIAO Fanghui. New Results on the Equivalence of Bivariate Polynomial Matrices[J]. Journal of Systems Science and Complexity, 2023, 36(1): 7795.
[1] Bose N K, Applied Multidimensional Systems Theory, Van Nostrand Reinhold, New York, 1982. [2] Bose N K, Buchberger B, and Guiver J, Multidimensional Systems Theory and Applications, Dordrecht, Kluwer, The Netherlands, 2003. [3] Kailath T, Linear Systems, Englewood Cliffs, NJ:Prentice Hall, 1980. [4] Rosenbrock H H, State Space and Multivariable Theory, NelsonWiley, New Work, London, 1970. [5] Morf M, Levy B, and Kung S, New results in 2D systems theory:Part I, Proceeding of the IEEE, 1977, 65:861872. [6] Frost M and Storey C, Equivalence of a matrix over R[s, z] with its Smith form, International Journal of Control, 1978, 28(5):665671. [7] Lee E and Zak S, Smith forms over R[z1, z2], IEEE Transactions on Automatic Control, 1983, 28(1):115118. [8] Frost M and Boudellioua M, Some further results concerning matrices with elements in a polynomial ring, International Journal of Control, 1986, 43(5):15431555. [9] Pugh A C, McInerney S J, and ElNabrawy E M O, Equivalence and reduction of 2D systems, IEEE Transactions on Circuits and Systems II:Express Briefs, 2005, 52(5):271275. [10] Cluzeau T and Quadrat A, Factoring and decomposing a class of linear functional systems, Linear Algebra and Its Applications, 2008, 428:324381. [11] Boudellioua M S and Quadrat A, Serre's reduction of linear function systems, Mathematics in Computer Science, 2010, 4(23):289312. [12] Cluzeau T and Quadrat A, A new insight into Serre's reduction problem, Linear Algebra and its Applications, 2015, 483:40100. [13] Lin Z, Boudellioua M S, and Xu L, On the equivalence and factorization of multivariate polynomial matrices, Proceeding of ISCAS, Greece, 2006, 49114914. [14] Li D, Liu J, and Zheng L, On the equivalence of multivariate polynomial matrices, Multidimensional Systems and Signal Processing, 2017, 28(1):225235. [15] Li D and Liang R, Serre's reduction and the Smith forms of multivariate polynomial matrices, Mathematical Problems in Engineering, 2020, 113, DOI:10.1155/2020/5430842. [16] Li D, Liu J, and Chu D, The Smith form of a multivariate polynomial matrix over an arbitrary coefficient field, Linear and Multilinear Algebra, 2022, 70(2):366379. [17] Li D, Liu J, and Zheng L, On serre reduction of multidimensional systems, Mathematical Problems in Engineering, 2020, 18, DOI:10.1155/2020/7435237. [18] Lu D, Wang D, and Xiao F, Further results on the factorization and equivalence for multivariate polynomial matrices, Proceedings of the 45th International Symposium on Symbolic and Algebraic Computation, 2020, 328335. [19] Li D, Liang R, and Liu J, Some further results on the Smith form of bivariate polynomial matrices, Journal of System Science and Mathematical Science (Chinese Series), 2019, 39(12):19831997. [20] Strang G, Linear Algebra and Its Applications, Academic Press, Cambridge, Massachusetts, 2010. [21] Youla D and Gnavi G, Notes on ndimensional system theory, IEEE Transactions on Circuits and Systems, 1979, 26(2):105111. [22] Quillen D, Projective modules over polynomial rings, Inventiones Mathematicae, 1976, 36(1):167171. [23] Suslin A A, Projective modules over polynomial rings are free, Soviet Mathematics Doklady, 1976, 17:11601164. [24] Serre J P, Faisceaux algébriques cohérents, Annals of Mathematics, 1955, 61(2):197278. [25] Logar A and Sturmfels B, Algorithms for the QuillenSuslin theorem, Journal of Algebra, 1992, 145(1):231239. [26] Park H, A computational theory of laurent polynomial rings and multidimensional FIR systems, Doctoral Dissertation, University of California at Berkeley, USA, 1995. [27] Youla D and Pickel P, The QuillenSuslin theorem and the structure of ndimensional elementary polynomial matrices, IEEE Transactions on Circuits and Systems, 1984, 31(6):513518. [28] Fabiá nska A and Quadrat A, Applications of the QuillenSuslin theorem to multidimensional systems theory, Eds. by Park H, Regensburger G, Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computational and Applied Mathematics, Walter de Gruyter, 2007, 3:23106. [29] Lin Z, On matrix fraction descriptions of multivariable linear nD systems, IEEE Transactions on Circuits and Systems, 1988, 35(10):13171322. [30] Lin Z and Bose N, A generalization of serre's conjecture and some related issues, Linear Algebra and Its Applications, 2001, 338:125138. [31] Pommaret J F, Solving bose conjecture on linear multidimensional systems, Proceedings of European Control Conference, IEEE, Porto, Portugal, 2001, 16531655. [32] Wang M and Feng D, On LinBose problem, Linear Algebra and Its Applications, 2004, 390(1):279285. [33] Pugh A C, McInerney S J, and ElNabrawy E M O, Zero structures of nD systems, International Journal of Control, 2005, 78(4):277285. [34] Li L, Li X, and Lin Z, Stability and stabilisation of linear multidimensional discrete systems in the frequency domain, International Journal of Control, 2013, 86(11):19691989. 
[1]  LIU Jinwang, WU Tao, LI Dongmei. Smith Form of Triangular Multivariate Polynomial Matrix [J]. Journal of Systems Science and Complexity, 2023, 36(1): 151164. 
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