Linear recurrent sequences are those whose elements are defined as linear combinations of preceding elements, and finding recurrence relations 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. Finding the ideal of their recurrence relations has applications such as the computation of minimal polynomials and determinants of sparse matrices over $\mathbb{A}$. We present three methods for finding this ideal: 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 relations and by exploiting the Hankel structure to compress the approximation problem. Then we confirm these improvements empirically through a C++ implementation, and we discuss the above-mentioned applications.

翻译：线性重复序列是指其元素被定义为前元素的线性组合,而发现重复关系是计算机代数中一个根本问题。 在本文中,我们侧重于其元素是环的矢量的序列 $\ mathbb{A}=\ mathb{K}K}[x]/(xx*d)$(truced monnocial) $。 查找其重现关系的理想有各种应用,例如计算最小的多元和稀薄基质的决定因素$\mathbb{A}。 我们提出了找到这一理想的三种方法: 一种属于Kurakin的Berlekamp- Masssey- 类似方法, 一种通过最小的近似 $\ mathb{A} 来计算某块- Hankel 矩阵的内核, 以及一种基于双向帕德近似值的方法。 我们建议前两种方法的复杂度改进方法, 分别避免计算冗余关系, 利用 和 Hankel 结构来压缩近似近似问题。 我们随后通过 C+ 应用程序确认这些改进。