Victor Tapia

Departamento de Matemáticas Universidad Nacional de Colombia Bogotá, Colombia

e-mail: tapiens@gmail.com

Abstract. We develop an algorithm to construct algebraic invariants for hyper-matrices. We then construct hyper-determinants and exhibit a generalization of the Cayley-Hamilton theorem for hyper-matrices.

Keywords and phrases. Polynomial identities, hyper-matrices.

2000 Mathematics Subject Classification. Primary: 14M12. Secondary: 15A24.

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[1] A. Acín, J.I. Latorre & P. Pascual, Three-party entanglement from positronium, Phys. Rev. A. 63, 042107 (2001).         [ Links ]

[2] D. G. Antzoulatos & A. A. Sawchuk, Hypermatrix algebra: Theory, CVGIP: Image Understanding 57 (1993), 24-41.         [ Links ]

[3] D. G. Antzoulatos & A. A. Sawchuk, Hypermatrix algebra: Applications in parallel imaging processing, CVGIP: Image Understanding 57 (1993), 42-62.         [ Links ]

[4] G. S. Asanov, Finsler Geometry, Relativity and Gauge Theories, Reidel, Dordrecht, 1985.         [ Links ]

[5] E. Briand, J.-G. Luque & J.-Y. Thibon, A complete set of covariants of the four qubit system, J. Phys. A: Math. Gen. 36 (2003), 9915-9927.        [ Links ]

[6] A. Cayley, On the Theory of Linear Transformations, Cambridge Math. J. 4 (1845), 193-209.         [ Links ]

[7] V. Coffman, J. Kundu & W.K. Wootters, Distributed entanglement, Phys. Rev. A. 61 052306 (2000).         [ Links ]

[8] I. M. Gelfand, M. M. Kapranov & A. V. Zelevinsky, Hyperdeterminants, Adv. Math. 96 (1992), 226-263.         [ Links ]

[9] I. M. Gelfand, M. M. Kapranov & A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants, BirkhÄauser, Boston, 1994.         [ Links ]

[10] C. M. Hull, Strongly coupled gravity and duality, Nucl. Phys. 583 B (2000), 237-259.         [ Links ]

[11] C. M. Hull, Symetries and compactifications of (4; 0) conformal gravity, J. High Energy Phys. 12 007 (2000).         [ Links ]

[12] C. M. Hull, Duality in gravity and higher spin gauge fields, J. High Energy Phys. 9 027 (2001).         [ Links ]

[13] J.-G. Luque & J.-Y. Thibon, Polynomial invariants of four qubits, Phys. Rev. A. 67 042303 (2003).         [ Links ]

[14] P. A. MacMahon, Combinatory Analysis (Cambridge University Press, 1915); reprinting Chelsea, New York, 1960.        [ Links ]

[15] H. Rund, The Differential Geometry of Finsler Spaces Springer, Berlin, 1959.         [ Links ]

[16] M. L. Stein & P.R. Stein, Enumeration of stochastic matrices with integer elements, Los Alamos Scientific Laboratory, report LA-4434 (1970).         [ Links ]

[17] R. P. Stanley, Linear homogeneous diophantine equations and magic labelings of graphs, Duke Math. J. 40 (1973), 607-632.         [ Links ]

[18] V. Tapia, Integrable conformal field theory in four dimensions and fourth-rank geometry, Int. J. Mod. Phys. D. 3(1993), 413-429.        [ Links ]

[19] V. Tapia, D. K. Ross, A. L. Marrakchi & M. Cataldo, Renormalizable conformally invariant model for the gravitational field, Class. Quantum Grav. 13 (1996), 3261-3267.         [ Links ]

[20] V. Tapia & D. K. Ross, Conformal fourth-rank gravity, non-vanishing cosmo- logical constant, and anisotropy, Class. Quantum Grav. 15 (1998), 245-249.         [ Links ]

[21] V. Tapia, Polynomial identities for hypermatrices, math-ph/0208010 (2002).         [ Links ]

[22] J. Weyman, Calculating discriminants by higher direct images, Trans. Am. Math. Soc. 343 (1994), 367-389.         [ Links ]

[23] J. Weyman & A. V. Zelevinsky, Singularities of hyperdeterminants, Ann. Inst. Fourier Grenoble 46 (1996), 591-644.         [ Links ]

[24] J. Weyman & A. V. Zelevinsky, Multiplicative properties of projectively dual varieties, Manuscripta Math. 82 (1994), 139-148.        [ Links ]

(Recibido en mayo de 2005. Aceptado en agosto de 2005)