Polynomial identities for ternary intermolecular recombination

TL;DR

This paper investigates polynomial identities satisfied by a ternary recombination operation inspired by DNA computing, using computer algebra to analyze identities up to degree 9 and comparing computational methods.

Contribution

It introduces a systematic computational approach to identify polynomial identities for ternary recombination operations and compares two algebraic methods for large matrix nullspace calculations.

Findings

Identified polynomial identities of degree up to 9 for the ternary operation.

Compared effectiveness of row canonical form and Hermite normal form methods.

Formulated conjectures for general n-ary recombination identities.

Abstract

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 <= 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: (i) the row canonical form, and (ii) the Hermite normal form with lattice basis reduction. In the conclusion, we formulate some conjectures for the general case of n-ary intermolecular recombination.