Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

Berthomieu, Jérémy; Bostan, Alin; Ferguson, Andrew; Din, Mohab Safey El

doi:10.1016/j.jalgebra.2022.03.002

Cited by 2 publications

(8 citation statements)

References 25 publications

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“…With the above perspective we expect the change of order step to be consistently faster than the F 4 -tracer step. Furthermore, even when computing over a 30-bit prime field, with only F 4 and no tracer, FGLM could take more than 25% of the total time, sometimes even close to 40% (see rand (4,7) or (4, 8)), now it is closer to negligible (often below or close to 10%).…”

Section: Practical Performancementioning

confidence: 99%

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

Berthomieu

Neiger

Din

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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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.

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Section: Practical Performancementioning

confidence: 99%

“…Compared to the previous 𝑂 (𝑡𝐷 2 ), the speed-up factor is of the order of 𝑡 2−𝜔 𝐷. We give explicit complexity gains for families of polynomial systems for which closed formulas or asymptotic estimates for 𝑡 and 𝐷 are known [7,18].…”

Section: Introductionmentioning

confidence: 99%

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

Berthomieu

Neiger

Din

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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“…For an 𝑛 × 𝑝, with 𝑛 ≤ 𝑝, general variable matrix, the authors of [5] simplify a formula given in [11] for the Hilbert series of the ideal generated by its maximal minors. The reduced numerator in this simplified formula of the Hilbert series,…”

Section: Preliminariesmentioning

confidence: 99%

“…Inspired by [19], there have been attempts to study the complexity of the Sparse-FGLM algorithm for systems with special structures, the main task being to estimate the sparsity of the multiplication matrices involved. Research in this direction was undertaken in [5]. Focusing on zero-dimensional ideals defining critical loci of polynomial maps restricted to algebraic sets, [5] introduces an explicit formula of the Hilbert series of those given ideals which significantly simplifies the formula given in [11].…”

Section: Introductionmentioning

confidence: 99%

“…Research in this direction was undertaken in [5]. Focusing on zero-dimensional ideals defining critical loci of polynomial maps restricted to algebraic sets, [5] introduces an explicit formula of the Hilbert series of those given ideals which significantly simplifies the formula given in [11]. This allows one to derive a sharp asymptotic bound for the number of non-zero entries of the multiplication matrix 𝑇 𝑥 𝑘 , when the number of variables 𝑘 tends to infinity.…”

Section: Introductionmentioning

confidence: 99%

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

Ferguson

2022

Proceedings of the 2022 International Symposium on Symbolic and Algebraic Computation

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Polynomial matrices and ideals generated by their minors appear in various domains such as cryptography, polynomial optimization and effective algebraic geometry. When the given matrix is symmetric, this additional structure on top of the determinantal structure, affects computations on the derived ideals. Thus, understanding the complexity of these computations is important. Moreover, this study serves as a stepping stone towards further understanding the effects of structure in determinantal systems, such as those coming from moment matrices. In this paper, we focus on the Sparse-FGLM algorithm, the state-of-the-art for changing ordering of Gröbner bases of zero-dimensional ideals. Under a variant of Fröberg's conjecture, we study its complexity for symmetric determinantal ideals and identify the gain of exploiting sparsity in the Sparse-FGLM algorithm compared with the classical FGLM algorithm. For an 𝑛 × 𝑛 symmetric matrix with polynomial entries of degree 𝑑, we show that the complexity of Sparse-FGLM for zero-dimensional determinantal ideals obtained from this matrix over that of the FGLM algorithm is at least 𝑂 (1/𝑑). Moreover, for some specific sizes of minors, we prove finer results of at least 𝑂 (1/𝑛𝑑) and 𝑂 (1/𝑛 3 𝑑).

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Gröbner bases and critical values: The asymptotic combinatorics of determinantal systems

Cited by 2 publications

References 25 publications

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

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

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

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