An application of lattice basis reduction to polynomial identities for algebraic structures
In this paper, the authors study computationally how to simplify the polynomial identities describing the trilineat operations classified by the authors in [Commun. Algebra 35, No. 9, 2932--2959 (2007; Zbl 1172.17003)]. The authors give a lattice basis reduction to simplify these identities equivalent to the ones obtained considering row canonical forms. To do so, the polynomial identities are represented as the nullspace of a large matrix \(E\) with entries in \(\mathbb Z\) (the expansion matrix). From the row canonical form of \(E\) over \(\mathbb Q\), an integral basis for the nullspace is extracted as a vector space over \(\mathbb Q\). Next they obtain a basis for the nullspace lattice by using the Hermite normal form of the transpose \(E^t\) and a matrix \(U\) such that \(U\cdot E^t=H\). If \(n\) is the dimension of the nullspace of \(E\) over \(\mathbb Q\), the last \(n\) rows of \(U\) form an integral basis for the nullspace lattice. To simplify these basis vectors, the LLL algorithm is used. The paper applies this approach for obtaining simpler polynomial identities for the most difficult case in the classification of trilinear operations. The authors show the pseudo-code of several algorithms applied to compute the lattice basis reduction.
scientific article; zbMATH DE number 5382406
scientific article