Geodesic
Triangularization of Matrices and Polynomial Maps
Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 94-105

Voir la notice de l'article provenant de la source Cambridge University Press

We present conditions for a set of matrices satisfying a permutation identity to be simultaneously triangularizable. As applications of our results, we generalize Radjavi’s result on triangularization of matrices with permutable trace and results by Yan and Tang on linear triangularization of polynomial maps.
DOI : 10.4153/S0008439519000250
Mots-clés : matrix, Jacobian matrix, permutation, polynomial map, simultaneous triangularization 
Li, Yueyue; Tian, Yan; Du, Xiankun. Triangularization of Matrices and Polynomial Maps. Canadian mathematical bulletin, Tome 63 (2020) no. 1, pp. 94-105. doi: 10.4153/S0008439519000250
@article{10_4153_S0008439519000250,
     author = {Li, Yueyue and Tian, Yan and Du, Xiankun},
     title = {Triangularization of {Matrices} and {Polynomial} {Maps}},
     journal = {Canadian mathematical bulletin},
     pages = {94--105},
     year = {2020},
     volume = {63},
     number = {1},
     doi = {10.4153/S0008439519000250},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000250/}
}
TY  - JOUR
AU  - Li, Yueyue
AU  - Tian, Yan
AU  - Du, Xiankun
TI  - Triangularization of Matrices and Polynomial Maps
JO  - Canadian mathematical bulletin
PY  - 2020
SP  - 94
EP  - 105
VL  - 63
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000250/
DO  - 10.4153/S0008439519000250
ID  - 10_4153_S0008439519000250
ER  - 
%0 Journal Article
%A Li, Yueyue
%A Tian, Yan
%A Du, Xiankun
%T Triangularization of Matrices and Polynomial Maps
%J Canadian mathematical bulletin
%D 2020
%P 94-105
%V 63
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/S0008439519000250/
%R 10.4153/S0008439519000250
%F 10_4153_S0008439519000250

[1] Bass, H., Connell, E. H., and Wright, D., The Jacobian Conjecture: Reduction of degree and formal expansion of the inverse. Bull. Amer. Math. Soc. 7(1982), no. 2, 287–330. https://doi.org/10.1090/S0273-0979-1982-15032-7 Google Scholar

[2] Cheng, C. C., Cubic linear Keller maps. J. Pure Appl. Algebra 160(2001), no. 1, 13–19. https://doi.org/10.1016/S0022-4049(00)00076-1 Google Scholar

[3] De Bondt, M., The strong nilpotency index of a matrix. Linear Multilinear Algebra 62(2014), no. 4, 486–497. https://doi.org/10.1080/03081087.2013.784282 Google Scholar

[4] De Bondt, M. and Van Den Essen, A., The Jacobian Conjecture: Linear triangularization for homogeneous polynomial maps in dimension three. J. Algebra 294(2005), no. 1, 294–306. https://doi.org/10.1016/j.jalgebra.2005.04.018 Google Scholar

[5] Dehghan, M. A. and Radjabalipour, M., Matrix algebras and Radjavi’s trace condition. Linear Algebra Appl. 148(1991), 19–25. https://doi.org/10.1016/0024-3795(91)90083-9 Google Scholar

[6] Dixon, J. D. and Mortimer, B, Permutation groups. Graduate Texts in Mathematics, 163, Springer-Verlag, New York, 1996. https://doi.org/10.1007/978-1-4612-0731-3 Google Scholar

[7] Guo, H., De Bondt, M., Du, X., and Sun, X., Polynomial maps with invertible sums of Jacobian matrices and directional derivatives. Indag. Math. 23(2012), no. 3, 256–268. https://doi.org/10.1016/j.indag.2011.11.007 Google Scholar

[8] Hadwin, D., Radjavi’s trace condition for triangularizability. J. Algebra 109(1987), no. 1, 184–192. https://doi.org/10.1016/0021-8693(87)90172-4 Google Scholar

[9] Liu, D., Du, X., and Sun, X., Quadratic linear Keller maps of nilpotency index three. Linear Algebra Appl. 429(2008), no. 1, 12–17. https://doi.org/10.1016/j.laa.2008.01.032 Google Scholar

[10] Meisters, G. H. and Olech, C., Strong nilpotence holds in dimensions up to five only. Linear Multilinear Algebra 30(1991), no. 2, 231–255. https://doi.org/10.1080/03081089108818109 Google Scholar

[11] Pate, K. and Cheng, C. C., Quadratic homogeneous Keller maps of rank two. Linear Algebra Appl. 476(2015), 16–27. https://doi.org/10.1016/j.laa.2015.02.014 Google Scholar

[12] Putcha, M. S. and Yaqub, A., Semigroups satisfying permutation identities. Semigroup Forum 3(1971), no. 1, 68–73. https://doi.org/10.1007/BF02572944 Google Scholar

[13] Radjavi, H., A trace condition equivalent to simultaneous triangularizability. Canad. J. Math. 38(1986), no. 2, 376–386. https://doi.org/10.4153/CJM-1986-018-1 Google Scholar

[14] Radjavi, H. and Rosenthal, P., Simultaneous triangularization. Springer-Verlag, New York, 2000. https://doi.org/10.1007/978-1-4612-1200-3 Google Scholar

[15] Rusek, K., Linearly triangularizable quadratic endomorphisms. Linear Multilinear Algebra 64(2016), no. 8, 1527–1537. https://doi.org/10.1080/03081087.2015.1099601 Google Scholar

[16] Sun, X., On additive-nilpotency of Jacobian matrices of polynomial maps. Linear Algebra Appl. 439(2013), no. 12, 3746–3751. https://doi.org/10.1016/j.laa.2013.10.014 Google Scholar

[17] Sun, X., On the strong nilpotence problem. Algebra Colloq. 21(2014), no. 1, 117–128. https://doi.org/10.1142/S100538671400008X Google Scholar

[18] Van Den Essen, A., Polynomial automorphisms and the Jacobian conjecture. Progress in Mathematics, 190, Birkhäuser, Berlin, 2000. https://doi.org/10.1007/978-3-0348-8440-2 Google Scholar

[19] Van Den Essen, A. and Hubbers, E., Polynomial maps with strongly nilpotent Jacobian matrix and the Jacobian conjecture. Linear Algebra Appl. 247(1996), no. 6, 121–132. https://doi.org/10.1016/0024-3795(95)00095-X Google Scholar

[20] Watters, J. F., Block triangularization of algebras of matrices. Linear Algebra Appl. 32(1980), 3–7. https://doi.org/10.1016/0024-3795(80)90003-8 Google Scholar

[21] Yan, D. and Tang, G., The linear triangularizability of some Keller maps. Linear Algebra Appl. 438(2013), no. 9, 3649–3653. https://doi.org/10.1016/j.laa.2012.12.018 Google Scholar

[22] Yu, Jie-Tai, Bass, H., Connell, E. H., and Wright, D., On generalized strongly nilpotent matrices. Linear Multilinear Algebra 41(1996), no. 1, 19–22. https://doi.org/10.1080/03081089608818457 Google Scholar

Cité par Sources :