Computing submatrices of the Hermite normal form of a structured polynomial matrix

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.

💡 Research Summary

The paper addresses the problem of computing only a leading principal submatrix of the Hermite normal form (HNF) of a univariate polynomial matrix that possesses a small displacement rank, a situation that frequently arises in applications such as bivariate resultant computation and Gröbner basis change of ordering. While recent advances have shown that the full HNF of an n × n polynomial matrix can be obtained in essentially the same cost as polynomial matrix multiplication (˜O(n^ω·deg M)), these algorithms do not exploit the underlying displacement structure and therefore remain costly when only a small m × m submatrix (m ≪ n) is needed.

The authors propose a novel algorithmic framework that combines evaluation–interpolation with fast Las‑Vegas randomized structured linear algebra. The key observation is that, given displacement generators S,T∈K