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 $\mathbb{A}$.

中文翻译：

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多项式线性递归序列的算法

线性递归序列是那些将其元素定义为先前元素的线性组合的序列，并且找到递归关系是计算机代数中的一个基本问题。在本文中，我们关注于其元素为截断多项式的环\\ mathbb {A} = \ mathbb {K} [x] /（x ^ d）$上的向量的序列。我们提供了三种寻找消除多项式理想的方法：一种归因于Kurakin的类似于Berlekamp-Massey的方法，一种是通过最小逼近来计算$ \ mathbb {A} $上某些Block-Hankel矩阵的核，另一种是基于二元Pad \'e近似值。我们分别针对前两种方法提出了复杂度改进措施，分别是避免了冗余序列发生器的计算以及通过利用汉克尔结构来压缩近似实例。然后，我们通过C ++实现在经验结果中观察到这些改进。最后，我们讨论了$ \ mathbb {A} $上最小多项式和稀疏矩阵行列式的计算应用。