Faster change of order algorithm for Gr\"obner bases under shape and stability assumptions

TL;DR

This paper introduces a faster algorithm for changing the order of Gr"obner bases in zero-dimensional polynomial systems, exploiting shape and stability properties to improve computational complexity and practical performance.

Contribution

It presents a novel change of order algorithm leveraging algebraic structure and matrix normal forms, reducing complexity under shape and stability assumptions.

Findings

The new algorithm has complexity $O\tilde{~}(t^{\omega-1}D)$, improving over previous bounds.

Practical experiments show significant performance gains.

The method effectively exploits the structure of the multiplication matrix.

Abstract

Solving zero-dimensional polynomial systems using Gr\"obner bases is usually done by, first, computing a Gr\"obner basis for the degree reverse lexicographic order, and next computing the lexicographic Gr\"obner basis with a change of order algorithm. Currently, the change of order now takes a significant part of the whole solving time for many generic instances. Like the fastest known change of order algorithms, this work focuses on the situation where the ideal defined by the system satisfies natural properties which can be recovered in generic coordinates. First, the ideal has a \emph{shape} lexicographic Gr\"obner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a \emph{stability} property; in particular, the multiplication matrix can be read on the input Gr\"obner basis. The current fastest algorithms rely on the sparsity of this matrix. Actually, this sparsity is a consequence of an algebraic structure, which can be exploited to represent the matrix concisely as a univariate polynomial matrix. We show that the Hermite normal form of that matrix yields the sought lexicographic Gr\"obner basis, under assumptions which cover the shape position case. Under some mild assumption implying $n \le t$, the arithmetic complexity of our algorithm is $O\tilde{~}(t^{\omega-1}D)$, where $n$ is the number of variables, $t$ is a sparsity indicator of the aforementioned matrix, $D$ is the degree of the zero-dimensional ideal under consideration, and $\omega$ is the exponent of matrix multiplication. This improves upon both state-of-the-art complexity bounds $O\tilde{~}(tD^2)$ and $O\tilde{~}(D^\omega)$, since $\omega < 3$ and $t\le D$. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit.