A dynamic programming method for building free algebras

AbstractWe are interested in deciding if a given nonassociative polynomial h is an identity for a variety of nonassociative algebras. We present an algorithm for constructing a certain homomorphic image of a free nonassociative algebra which is sufficient to answer the question. The algorithm resembles dynamic programming in that the algebra is built by constructing a sequence of subspaces; the basis of each subspace is determined by the basis of previous subspaces. The number of arithmetic operations required to construct the algebra is bounded by a polynomial in the degree of h and the dimension of the resulting algebra

Reducing the adjacency matrix of a tree

Let T be a tree, A its adjacency matrix, and a scalar. We describe a linear-time algorithm for reducing the matrix In + A. Applications include computing the rank of A, nding a maximum matching in T , computing the rank and determinant of the associated neighborhood matrix, and computing the characteristic polynomial of A