Algorithms for Linearly Recurrent Sequences of Truncated Polynomials

Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding relations of recurrences is a fundamental problem in computer algebra. In this paper, we focus on sequences whose elements are vectors over the ring $\mathbb{A} = \mathbb{K}[x]/(x^d)$ of truncated polynomials. We present three methods for finding the ideal of canceling polynomials: a Berlekamp-Massey-like approach due to Kurakin, one which computes the kernel of some block-Hankel matrix over $\mathbb{A}$ via a minimal approximant basis, and one based on bivariate Pad\'e approximation. We propose complexity improvements for the first two methods, respectively by avoiding the computation of redundant sequence generators and by exploiting the Hankel structure to compress the approximation instance. We then observe these improvements in our empirical results through a C++ implementation. Finally we discuss applications to the computation of minimal polynomials and determinants of sparse matrices over $\mathbb{A}$.