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.

翻译：经过数十年的连续算法改进，2010年代的研究表明，单变量多项式矩阵的埃尔米特标准形（HNF）可在本质上等同于多项式矩阵乘法复杂度的时间界内完成计算。近期，关于双变量多项式与Gröbner基的若干研究凸显了计算具有小位移秩的结构化多项式矩阵的行列式或HNF的重要性。在此类情形下，HNF的一个小型前主子矩阵往往包含全部所需信息。本文展示了如何利用位移结构加速此类子矩阵的计算。为实现这一目标，我们借助求值-插值技术，在域上采用结构化线性代数方法。这使得我们能从输入矩阵的逆中恢复若干行，进而通过关系基推导出目标HNF子矩阵。