Lu, Dong; Wang, Dingkang; Xiao, Fanghui New remarks on the factorization and equivalence problems for a class of multivariate polynomial matrices. (English) Zbl 1497.15017 J. Symb. Comput. 115, 266-284 (2023). Summary: This paper is concerned with the factorization and equivalence problems of multivariate polynomial matrices. We present some new criteria for the existence of matrix factorizations for a class of multivariate polynomial matrices, and obtain a necessary and sufficient condition for the equivalence of a square polynomial matrix and a diagonal matrix. Based on the constructive proof of the new criteria, we give a factorization algorithm and prove the uniqueness of the factorization. We implement the algorithm on Maple, and two illustrative examples are given to show the effectiveness of the algorithm. MSC: 15A23 Factorization of matrices 13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 68W30 Symbolic computation and algebraic computation Keywords:multivariate polynomial matrices; matrix factorization; matrix equivalence; column reduced minors; Gröbner basis Software:SINGULAR; Maple PDFBibTeX XMLCite \textit{D. Lu} et al., J. Symb. Comput. 115, 266--284 (2023; Zbl 1497.15017) Full Text: DOI arXiv References: [1] Bose, N., Applied Multidimensional Systems Theory (1982), Van Nostrand Reinhold Co.: Van Nostrand Reinhold Co. New York · Zbl 0574.93031 [2] Bose, N.; Buchberger, B.; Guiver, J., Multidimensional Systems Theory and Applications (2003), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, The Netherlands · Zbl 1046.93001 [3] Boudellioua, M., Computation of the Smith form for multivariate polynomial matrices using Maple, Am. J. Comput. Math., 2, 1, 21-26 (2012) [4] Boudellioua, M., Further results on the equivalence to Smith form of multivariate polynomial matrices, Control Cybern., 42, 2, 543-551 (2013) · Zbl 1318.93044 [5] Boudellioua, M., Computation of a canonical form for linear 2-D systems, Int. J. Comput. Math., 2014, 487465, 1-6 (2014) [6] Boudellioua, M.; Quadrat, A., Serre’s reduction of linear function systems, Math. Comput. Sci., 4, 2-3, 289-312 (2010) · Zbl 1275.16003 [7] Charoenlarpnopparut, C.; Bose, N., Multidimensional FIR filter bank design using Gröbner bases, IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., 46, 12, 1475-1486 (1999) · Zbl 1075.94513 [8] Cluzeau, T.; Quadrat, A., Factoring and decomposing a class of linear functional systems, Linear Algebra Appl., 428, 324-381 (2008) · Zbl 1131.15011 [9] Cluzeau, T.; Quadrat, A., Isomorphisms and Serre’s reduction of linear systems, (Proceedings of the 8th International Workshop on Multidimensional Systems (2013), VDE: VDE Erlangen, Germany), 1-6 [10] Cluzeau, T.; Quadrat, A., A new insight into Serre’s reduction problem, Linear Algebra Appl., 483, 40-100 (2015) · Zbl 1318.93029 [11] Cox, D.; Little, J.; O’shea, D., Using Algebraic Geometry, Graduate Texts in Mathematics (2005), Springer: Springer New York · Zbl 1079.13017 [12] Eisenbud, D., Commutative Algebra: With a View Toward Algebraic Geometry (2013), Springer: Springer New York [13] Fabiańska, A.; Quadrat, A., Applications of the Quillen-Suslin theorem to multidimensional systems theory, (Park, H.; Regensburger, G., Gröbner Bases in Control Theory and Signal Processing. Gröbner Bases in Control Theory and Signal Processing, Radon Series on Computational and Applied Mathematics, vol. 3 (2007), Walter de Gruyter) · Zbl 1197.13011 [14] Frost, M.; Boudellioua, M., Some further results concerning matrices with elements in a polynomial ring, Int. J. Control, 43, 5, 1543-1555 (1986) · Zbl 0587.15015 [15] Frost, M.; Storey, C., Equivalence of a matrix over R \([s, z]\) with its Smith form, Int. J. Control, 28, 5, 665-671 (1978) · Zbl 0394.93013 [16] Greuel, G.; Pfister, G., A SINGULAR Introduction to Commutative Algebra (2002), Springer-Verlag · Zbl 1023.13001 [17] Guan, J.; Li, W.; Ouyang, B., On rank factorizations and factor prime factorizations for multivariate polynomial matrices, J. Syst. Sci. Complex., 31, 6, 1647-1658 (2018) · Zbl 1445.15009 [18] Guan, J.; Li, W.; Ouyang, B., On minor prime factorizations for multivariate polynomial matrices, Multidimens. Syst. Signal Process., 30, 493-502 (2019) · Zbl 1430.15011 [19] Guiver, J.; Bose, N., Polynomial matrix primitive factorization over arbitrary coefficient field and related results, IEEE Trans. Circuits Syst., 29, 10, 649-657 (1982) · Zbl 0504.65020 [20] Kailath, T., Linear Systems (1993), Prentice Hall: Prentice Hall Englewood Cliffs, NJ [21] Kung, S.; Levy, B.; Morf, M.; Kailath, T., New results in 2-D systems theory, part II: 2-D state-space models-realization and the notions of controllability, observability, and minimality, (Proceedings of the IEEE, vol. 65 (1977)), 945-961 [22] Lee, E.; Zak, S., Smith forms over R \([ z_1, z_2]\), IEEE Trans. Autom. Control, 28, 1, 115-118 (1983) · Zbl 0511.93018 [23] Li, D.; Liu, J.; Zheng, L., On the equivalence of multivariate polynomial matrices, Multidimens. Syst. Signal Process., 28, 225-235 (2017) · Zbl 1373.15019 [24] Lin, Z., On matrix fraction descriptions of multivariable linear n-D systems, IEEE Trans. Circuits Syst., 35, 10, 1317-1322 (1988) · Zbl 0662.93036 [25] Lin, Z., On primitive factorizations for n-D polynomial matrices, (IEEE International Symposium on Circuits and Systems (1993)), 601-618 · Zbl 1024.93014 [26] Lin, Z., Notes on n-D polynomial matrix factorizations, Multidimens. Syst. Signal Process., 10, 4, 379-393 (1999) · Zbl 0939.93017 [27] Lin, Z., On syzygy modules for polynomial matrices, Linear Algebra Appl., 298, 1-3, 73-86 (1999) · Zbl 0983.15012 [28] Lin, Z., Further results on n-D polynomial matrix factorizations, Multidimens. Syst. Signal Process., 12, 2, 199-208 (2001) · Zbl 0996.93053 [29] Lin, Z.; Bose, N., A generalization of Serre’s conjecture and some related issues, Linear Algebra Appl., 338, 125-138 (2001) · Zbl 1017.13006 [30] Lin, Z.; Boudellioua, M.; Xu, L., On the equivalence and factorization of multivariate polynomial matrices, (Proceedings of ISCAS. Proceedings of ISCAS, Greece (2006)), 4911-4914 [31] Lin, Z.; Li, X.; Fan, H., On minor prime factorizations for n-D polynomial matrices, IEEE Trans. Circuits Syst. II, Express Briefs, 52, 9, 568-571 (2005) [32] Lin, Z.; Xu, L.; Bose, N., A tutorial on Gröbner bases with applications in signals and systems, IEEE Trans. Circuits Syst. I, Regul. Pap., 55, 1, 445-461 (2008) [33] Lin, Z.; Ying, J.; Xu, L., Factorizations for n-D polynomial matrices, Circuits Syst. Signal Process., 20, 6, 601-618 (2001) · Zbl 1024.93014 [34] Liu, J.; Li, D.; Wang, M., On general factorizations for n-D polynomial matrices, Circuits Syst. Signal Process., 30, 3, 553-566 (2011) · Zbl 1213.93034 [35] Liu, J.; Li, D.; Zheng, L., The Lin-Bose problem, IEEE Trans. Circuits Syst. II, Express Briefs, 61, 1, 41-43 (2014) [36] Liu, J.; Wang, M., Notes on factor prime factorizations for n-D polynomial matrices, Multidimens. Syst. Signal Process., 21, 1, 87-97 (2010) · Zbl 1298.93197 [37] Liu, J.; Wang, M., New results on multivariate polynomial matrix factorizations, Linear Algebra Appl., 438, 1, 87-95 (2013) · Zbl 1255.15017 [38] Liu, J.; Wang, M., Further remarks on multivariate polynomial matrix factorizations, Linear Algebra Appl., 465, 204-213 (2015) · Zbl 1310.15021 [39] Logar, A.; Sturmfels, B., Algorithms for the Quillen-Suslin theorem, J. Algebra, 145, 1, 231-239 (1992) · Zbl 0747.13020 [40] Lu, D.; Ma, X.; Wang, D., A new algorithm for general factorizations of multivariate polynomial matrices, (Proceedings of 42nd ISSAC (2017)), 277-284 · Zbl 1457.68328 [41] Lu, D.; Wang, D.; Xiao, F., Factorizations for a class of multivariate polynomial matrices, Multidimens. Syst. Signal Process., 31, 989-1004 (2020) · Zbl 1459.15014 [42] Lu, D.; Wang, D.; Xiao, F., Further results on the factorization and equivalence for multivariate polynomial matrices, (Proceedings of 45th ISSAC (2020)), 328-335 · Zbl 07300088 [43] Lu, D.; Wang, D.; Xiao, F., On factor left prime factorization problems for multivariate polynomial matrices, Multidimens. Syst. Signal Process. (2021) · Zbl 1473.15023 [44] Morf, M.; Levy, B.; Kung, S., New results in 2-D systems theory, part I: 2-D polynomial matrices, factorization, and coprimeness, (Proceedings of the IEEE, vol. 65 (1977)), 861-872 [45] Park, H., A computational theory of Laurent polynomial rings and multidimensional FIR systems (1995), University of California at Berkeley, Ph.D. thesis [46] Pommaret, J., Solving Bose conjecture on linear multidimensional systems, (Proceedings of European Control Conference (2001), IEEE), 1653-1655 [47] Pugh, A.; Mcinerney, S.; Boudellioua, M.; Johnson, D.; Hayton, G., A transformaition for 2-D linear systems and a generalization of a theorem of rosenbrock, Int. J. Control, 71, 3, 491-503 (1998) · Zbl 0987.93010 [48] Quillen, D., Projective modules over polynomial rings, Invent. Math., 36, 167-171 (1976) · Zbl 0337.13011 [49] Rosenbrock, H., State-Space and Multivariable Theory (1970), Nelson: Nelson London · Zbl 0246.93010 [50] Serre, J., Faisceaux algébriques cohérents, Ann. Math., 61, 2, 197-278 (1955) · Zbl 0067.16201 [51] Srinivas, V., A generalized Serre problem, J. Algebra, 278, 2, 621-627 (2004) · Zbl 1079.13005 [52] Strang, G., Linear Algebra and Its Applications (1980), Academic Press · Zbl 0463.15001 [53] Sule, V., Feedback stabilization over commutative rings: the matrix case, SIAM J. Control Optim., 32, 6, 1675-1695 (1994) · Zbl 0821.93025 [54] Suslin, A., Projective modules over polynomial rings are free, Sov. Math. Dokl., 17, 1160-1165 (1976) · Zbl 0354.13010 [55] Wang, M., On factor prime factorization for n-D polynomial matrices, IEEE Trans. Circuits Syst. I, Regul. Pap., 54, 6, 1398-1405 (2007) · Zbl 1374.15026 [56] Wang, M., Remarks on n-D polynomial matrix factorization problems, IEEE Trans. Circuits Syst. II, Express Briefs, 55, 1, 61-64 (2008) [57] Wang, M.; Feng, D., On Lin-Bose problem, Linear Algebra Appl., 390, 279-285 (2004) · Zbl 1056.15013 [58] Wang, M.; Kwong, C., On multivariate polynomial matrix factorization problems, Math. Control Signals Syst., 17, 4, 297-311 (2005) · Zbl 1098.93010 [59] Youla, D.; Gnavi, G., Notes on n-dimensional system theory, IEEE Trans. Circuits Syst., 26, 2, 105-111 (1979) · Zbl 0394.93004 [60] Youla, D.; Pickel, P., The Quillen-Suslin theorem and the structure of n-dimensional elementary polynomial matrices, IEEE Trans. Circuits Syst., 31, 6, 513-518 (1984) · Zbl 0553.13003 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.