Faster change of order algorithm for Gröbner bases under shape and stability assumptions
Résumé
Solving polynomial systems whose solution set is finite is usually done in two main steps: compute a Gröbner basis for the degree reverse lexicographic order, and perform a change of order to find the lexicographic Gröbner basis. The second step is generally considered as better understood, in terms of algorithms and complexity. Yet, after two decades of progress on the first step, it turns out that the change of order now takes a large part of the solving time for many instances, including those that are generic or reached after applying a random change of variables. Like the fastest known change of order algorithms, this work focuses on the latter situation, where the ideal defined by the system satisfies structural properties. First, the ideal has a shape lexicographic Gröbner basis. Second, the set of leading terms with respect to the degree reverse lexicographic order has a stability property; in particular, the multiplication matrix of the smallest variable is computed for free from the input Gröbner basis. The current fastest algorithms rely on the sparsity of this multiplication matrix to find its minimal polynomial efficiently using Wiedemann's approach. This paper starts from the observation that 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 Gröbner basis, under assumptions which cover the shape position case. This leads to an improved complexity bound for the second step. The practical benefit is also confirmed via implementations based on the state-of-the-art software libraries msolve and PML.
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