Polynomial identities for ternary intermolecular recombination

Abstract

The operation of binary intermolecular recombination, originating in the theory of DNAcomputing, permits a natural generalization to $n$-ary operations which performsimultaneous recombination of $n$ molecules. In the case $n = 3$, we use computeralgebra to determine the polynomial identities of degree $\le 9$ satisfied by this trilinearnonassociative operation. Our approach requires computing a basis for the nullspace ofa large integer matrix, and for this we compare two methods: the row canonical form, andthe Hermite normal form with lattice basis reduction. In the conclusion, we formulatesome conjectures for the general case of $n$-ary intermolecular recombination.