Finding a minimal polynomial vector set of a vector of nD arrays

Abstract

We propose an algorithm for finding efficiently a minimal set of correlated linear recurrences capable of generating a given vector of finite n-dimensional (nD) arrays. The output of the algorithm is a Gröbner basis of a module over the multivariate polynomial ring, provided that the size of the given arrays is sufficiently large in comparison with the degrees of the characteristic polynomials of the correlated linear recurrences found by the method. This algorithm is also an extension of the Berlekamp-Massey algorithm for finding a minimal polynomial set of an nD array. Although the algorithm has a close connection with the nD Berlekamp-Massey algorithm for multiple nD arrays, the former will find a minimal set of compound linear recurrences which relate all the nD arrays of the given vector while the latter finds a minimal set of linear recurrences which are in common to all the given nD arrays.