# 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 & Continuous Dynamical Systems - S, 2011, 4 (6) : 1387-1399. doi: 10.3934/dcdss.2011.4.1387
##### References:
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##### References:
 [1] M. R. Bremner, Jordan algebras arising from intermolecular recombination,, SIGSAM Bulletin (Communications in Computer Algebra), 39 (2005), 106.  doi: 10.1145/1140378.1140380.  Google Scholar [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.  doi: 10.1006/jabr.2000.8372.  Google Scholar [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.  doi: 10.1016/j.laa.2008.09.003.  Google Scholar [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.  doi: 10.1080/00927870802502886.  Google Scholar [5] L. Landweber and L. Kari, The evolution of cellular computing: nature's solution to a computational problem,, Biosystems, 52 (1999), 3.  doi: 10.1016/S0303-2647(99)00027-1.  Google Scholar [6] S. R. Sverchkov, Structure and representations of Jordan algebras arising from intermolecular recombination,, in, 483 (2009), 261.   Google Scholar [7] S. R. Sverchkov, Algebraic theory of DNA recombination,, Jordan Theory Preprint Archives (\url{http://homepage.uibk.ac.at/ c70202/jordan/}), (2009).   Google Scholar [8] I. M. Wanless, Permanents,, in, (2007).   Google Scholar
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