计算结构多项式矩阵的Hermite规范形的子矩阵
Computing submatrices of the Hermite normal form of a structured polynomial matrix
AI总结：
本文提出利用位移结构加速计算结构多项式矩阵Hermite规范形子矩阵的方法，通过评估-插值实现高效计算。
摘要：
在数十年的算法改进之后，2010年代的研究展示了如何在与多项式矩阵乘法复杂度相当的复杂度界内计算单变量多项式矩阵的Hermite规范形（HNF）。最近，关于双变量多项式和Gröbner基的研究突显了计算具有小位移秩的多项式矩阵的行列式或HNF的兴趣。在这种情况下，HNF的一个小的主子矩阵通常包含所有所需的信息。在本文中，我们展示了如何利用位移结构来加速此类子矩阵的计算。为此，我们依赖于通过评估-插值实现的结构线性代数。这使我们能够恢复输入矩阵的某些行，从而通过关系基推导出所需的HNF子矩阵。
英文摘要：
Following several decades of successive algorithmic improvements, works from the 2010s have showed how to compute the Hermite normal form (HNF) of a univariate polynomial matrix within a complexity bound which is essentially that of polynomial matrix multiplication. Recently, several results on bivariate polynomials and Gröbner bases have highlighted the interest of computing determinants or HNFs of polynomial matrices that happen to be structured, with a small displacement rank. In such contexts, a small leading principal submatrix of the HNF often contains all the sought information. In this article, we show how the displacement structure can be exploited in order to accelerate the computation of such submatrices. To achieve this, we rely on structured linear algebra over the field thanks to evaluation-interpolation. This allows us to recover some rows of the inverse of the input matrix, from which we deduce the sought HNF submatrix via bases of relations.
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