American Institute of Mathematical Sciences

December  2011, 4(6): 1387-1399. doi: 10.3934/dcdss.2011.4.1387

Polynomial identities for ternary intermolecular recombination

Received  March 2009 Revised  September 2009 Published  December 2010

The operation of binary intermolecular recombination, originating in the theory of DNA computing, permits a natural generalization to $n$-ary operations which perform simultaneous recombination of $n$ molecules. In the case $n = 3$, we use computer algebra to determine the polynomial identities of degree $\le 9$ satisfied by this trilinear nonassociative operation. Our approach requires computing a basis for the nullspace of a large integer matrix, and for this we compare two methods: the row canonical form, and the Hermite normal form with lattice basis reduction. In the conclusion, we formulate some conjectures for the general case of $n$-ary intermolecular recombination.
Citation: Murray R. Bremner. Polynomial identities for ternary intermolecular recombination. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1387-1399. doi: 10.3934/dcdss.2011.4.1387
References:
 [1] M. R. Bremner, Jordan algebras arising from intermolecular recombination, SIGSAM Bulletin (Communications in Computer Algebra), 39 (2005), 106-117. doi: 10.1145/1140378.1140380. [2] M. R. Bremner and I. R. Hentzel, Identities for generalized Lie and Jordan products on totally associative triple systems, Journal of Algebra, 231 (2000), 387-405. doi: 10.1006/jabr.2000.8372. [3] M. R. Bremner and L. A. Peresi, An application of lattice basis reduction to polynomial identitites for algebraic systems, Linear Algebra and its Applications, 430 (2009), 642-659. doi: 10.1016/j.laa.2008.09.003. [4] M. R. Bremner, Y. F. Piao and S. W. Richards, Polynomial identities for Bernstein algebras of simple Mendelian inheritance, Communications in Algebra, 37 (2009), 3438-3455. doi: 10.1080/00927870802502886. [5] L. Landweber and L. Kari, The evolution of cellular computing: nature's solution to a computational problem, Biosystems, 52 (1999), 3-13. doi: 10.1016/S0303-2647(99)00027-1. [6] S. R. Sverchkov, Structure and representations of Jordan algebras arising from intermolecular recombination, in "Algebras, Representations and Applications" (edited by V. Futorny, V. Kac, I. Kashuba and E. Zelmanov), Contemporary Mathematics, American Mathematical Society, 483 (2009), 261-285. [7] S. R. Sverchkov, Algebraic theory of DNA recombination, Jordan Theory Preprint Archives (http://homepage.uibk.ac.at/ c70202/jordan/), Preprint 278, 14 October 2009, 20 pages. [8] I. M. Wanless, Permanents, in "Handbook of Linear Algebra," (edited by L. Hogben), Chapman & Hall / CRC, 2007, Chapter 31.

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References:
 [1] M. R. Bremner, Jordan algebras arising from intermolecular recombination, SIGSAM Bulletin (Communications in Computer Algebra), 39 (2005), 106-117. doi: 10.1145/1140378.1140380. [2] M. R. Bremner and I. R. Hentzel, Identities for generalized Lie and Jordan products on totally associative triple systems, Journal of Algebra, 231 (2000), 387-405. doi: 10.1006/jabr.2000.8372. [3] M. R. Bremner and L. A. Peresi, An application of lattice basis reduction to polynomial identitites for algebraic systems, Linear Algebra and its Applications, 430 (2009), 642-659. doi: 10.1016/j.laa.2008.09.003. [4] M. R. Bremner, Y. F. Piao and S. W. Richards, Polynomial identities for Bernstein algebras of simple Mendelian inheritance, Communications in Algebra, 37 (2009), 3438-3455. doi: 10.1080/00927870802502886. [5] L. Landweber and L. Kari, The evolution of cellular computing: nature's solution to a computational problem, Biosystems, 52 (1999), 3-13. doi: 10.1016/S0303-2647(99)00027-1. [6] S. R. Sverchkov, Structure and representations of Jordan algebras arising from intermolecular recombination, in "Algebras, Representations and Applications" (edited by V. Futorny, V. Kac, I. Kashuba and E. Zelmanov), Contemporary Mathematics, American Mathematical Society, 483 (2009), 261-285. [7] S. R. Sverchkov, Algebraic theory of DNA recombination, Jordan Theory Preprint Archives (http://homepage.uibk.ac.at/ c70202/jordan/), Preprint 278, 14 October 2009, 20 pages. [8] I. M. Wanless, Permanents, in "Handbook of Linear Algebra," (edited by L. Hogben), Chapman & Hall / CRC, 2007, Chapter 31.
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