Faster Change of Order Algorithm for Gröbner Bases under Shape and Stability Assumptions

Abstract

Solving zero-dimensional polynomial systems using Gröbner bases is usually done by, first, computing a Gröbner basis for the degree reverse lexicographic order, and next computing the lexicographic Gröbner 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 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 can be read on the input Gröbner 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 Gröbner basis, under assumptions which cover the shape position case. Under some mild assumption implying 𝑛 ≤ 𝑡, the arithmetic complexity of our algorithm is 𝑂˜(𝑡 𝜔−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 𝜔 is the exponent of matrix multiplication. This improves upon both stateof-the-art complexity bounds 𝑂˜(𝑡𝐷 2 ) and 𝑂˜(𝐷 𝜔 ), since 𝜔 < 3 and 𝑡 ≤ 𝐷. Practical experiments, based on the libraries msolve and PML, confirm the high practical benefit. CCS CONCEPTS• Computing methodologies → Algebraic algorithms; • Theory of computation → Design and analysis of algorithms.

Finer Complexity Estimates for the Change of Ordering of Gröbner Bases for Generic Symmetric Determinantal Ideals

“…Our results describe the fundamental parameter 𝑚, the number of dense columns of 𝑇 𝑥 𝑘 . Therefore, while the complexity results in this article focus on the application to the Sparse-FGLM algorithm, we can also apply the propositions of Section 4 to the new changeof-ordering algorithm of [6]. There, the authors prove a complexity result, excluding logarithmic factors, of Õ (𝑚 𝜔 −1 𝐷), where 𝜔 is the exponent of the complexity of matrix multiplication.…”

“…Applying our estimates for 𝑚 leads to even finer complexity results for symmetric determinantal systems. Our bound on 𝑚 enables more precise comparison of this new algorithm in [6] with the existing algorithms based on fast linear algebra [17,32] whose complexities lie in Õ (𝐷 𝜔 ).…”

“…Comparing the Sparse-FGLM algorithm with these algorithms requires estimating the parameter 𝑚. Moreover, a bound on 𝑚 serves independently as an indicator for the sparsity of 𝑇 𝑥 𝑘 and could be useful for any algorithm that relies on this sparsity (e.g., the algorithm of [6] that improves [17,32]).…”

Bivariate polynomial reduction and elimination ideal over finite fields

“…3.10), and implement the operations on its elements. This direction has been taken in particular (Berthomieu et al, 2022b) for a change of ordering of Gröbner bases algorithm, and in (Schost and St-Pierre, 2023b) for a p-adic Gröbner basis algorithm.…”