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

TL;DR

This work addresses the efficient computation of a leading principal submatrix of the Hermite normal form (HNF) for structured polynomial matrices by exploiting small displacement rank through an evaluation-interpolation pipeline. The core idea is to recover rows of the inverse $M^{-1}$ modulo a modulus $A=\prod_i (x-a_i)$ via structured modular solves, and then obtain the desired HNF submatrix from a relation-basis construction of a module $\mathcal{M}_J$. The authors provide a versatile algorithm, HermiteFormSubmatrix, with randomized guarantees and complexity bounds that depend on the displacement rank $\alpha$, the matrix size $n$, and the target submatrix size $m$, including cases where generic column or row HNF properties hold. A key application is changing the monomial order for bivariate zero-dimensional ideals: by building a block-Toeplitz matrix $M$ from a $\preccurlyeq_{drl}$-Groebner basis, the lexicographic basis can be recovered from the leading $m\times m$ block of the HNF of $M$, under a shape-position assumption. Overall, the approach extends efficient determinant-like techniques for structured Sylvester-type matrices to a broad class of polynomial matrices and provides practical routes for lex- and shape-position basis extraction in Gröbner basis computations.

Abstract

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.

Figures (1)

• Figure 1: Representation of monomials in $\bar{\mathcal{G}}$ for an ideal $\mathcal{I}$ with a minimal $\mathrel{\preccurlyeq_{\operatorname{drl}}}$-Gröbner basis of $\ell=4$ elements $(g_0,g_1,g_2,g_3)$ whose leading monomials are $(x^7, x^4 y^5, x^3 y^8, y^{10})$. The greyed area is the $\mathrel{\preccurlyeq_{\operatorname{drl}}}$-monomial basis of $\mathbb{K}[x,y]/\mathcal{I}$; the dotted monomials are the above $\operatorname{lm}(g_i)$ for $0 \le i < \ell$; the horizontally striped ones are the other leading monomials of $\bar{\mathcal{G}}$, that is, $\operatorname{lm}(y^k g_i)$ for $1 \le k < n_i$ and $0 \le i < \ell$; and the diagonally striped ones are all other monomials that possibly appear in $\bar{\mathcal{G}}$. Here, $n = n_0 + n_1 + n_2 + n_3 = 12$ and $D = n_0 \mathrm{deg}{}_{x}(g_0) + n_1 \mathrm{deg}{}_{x}(g_1) + n_2 \mathrm{deg}{}_{x}(g_2) = 5\cdot 7 + 3 \cdot 4 + 2 \cdot 3 = 53$.

Theorems & Definitions (16)

• theorem 1
• theorem 2
• proposition 1
• proposition 2
• proposition 3
• proposition 4
• lemma 1
• lemma 2
• lemma 3
• lemma 4