[Paper Review] A probabilistic and deterministic modular algorithm for computing Groebner basis over $\Q$

Bernard Parisse|arXiv (Cornell University)|Sep 16, 2013

Polynomial and algebraic computation5 references3 citations

TL;DR

This paper presents a probabilistic and deterministic modular algorithm for computing Gröbner bases over ℚ using Chinese remaindering and Farey reconstruction, with a novel error-checking strategy that allows users to choose between fast probabilistic results and certified deterministic outcomes. The method achieves competitive performance—especially in memory efficiency and speed on multi-core systems—outperforming open-source tools like Singular and matching or surpassing commercial systems like Maple and Magma on certain benchmarks, while requiring only ~10K lines of code compared to highly optimized 200K-line systems.

ABSTRACT

Modular algorithm are widely used in computer algebra systems (CAS), for example to compute efficiently the gcd of multivariate polynomials. It is known to work to compute Groebner basis over $\Q$, but it does not seem to be popular among CAS implementers. In this paper, I will show how to check a candidate Groebner basis (obtained by reconstruction of several Groebner basis modulo distinct prime numbers) with a given error probability, that may be 0 if a certified Groebner basis is desired. This algorithm is now the default algorithm used by the Giac/Xcas computer algebra system with competitive timings, thanks to a trick that can accelerate computing Groebner basis modulo a prime once the computation has been done modulo another prime.

Motivation & Objective

• To address the lack of widespread adoption of modular algorithms for Gröbner basis computation over ℚ in computer algebra systems despite their theoretical advantages.
• To reduce the computational bottleneck of verifying reconstructed Gröbner bases by introducing a fast probabilistic checking method with user-controlled error probability.
• To improve efficiency through a learning technique that reuses information from prior modular computations to accelerate subsequent Gröbner basis calculations modulo different primes.
• To provide a practical, open-source implementation in Giac/Xcas that achieves competitive performance against both open-source and commercial systems.
• To enable both fast conjecture generation via probabilistic results and mathematically certified outcomes via deterministic checks.

Proposed method

• Reconstruct a candidate Gröbner basis over ℚ using Chinese remaindering on Gröbner bases computed modulo multiple distinct primes.
• Apply Farey reconstruction to recover rational coefficients from modular results, ensuring stability and correctness.
• Use a probabilistic checking step that verifies reduction to zero modulo several primes, with error probability bounded by ε.
• Implement a deterministic checking method that verifies reduction to zero over ℚ using quotient reconstruction via multiple primes and coefficient bounds.
• Leverage a 'learning' technique to accelerate modular Gröbner basis computation by reusing critical pair information from previous primes, reducing redundant work.
• Integrate the algorithm into Giac/Xcas with support for parallelization and configurable error tolerance.

Experimental results

Research questions

• RQ1 Can a modular algorithm for Gröbner basis computation over ℚ be made both efficient and practical for real-world use in open-source computer algebra systems?
• RQ2 How can the verification step—often the bottleneck—be accelerated without sacrificing correctness, especially when aiming for mathematical certification?
• RQ3 To what extent can learning from prior modular computations reduce redundant work in Gröbner basis algorithms, particularly in comparison to F5-style methods?
• RQ4 How does the performance of this modular approach compare to state-of-the-art implementations like Magma, Maple, and Singular on standard benchmarks such as Cyclic and Katsura?
• RQ5 Can a hybrid probabilistic-deterministic framework balance speed and correctness effectively for both conjecture generation and formal proof?

Key findings

• The deterministic modular algorithm in Giac/Xcas outperforms Singular on Cyclic and Katsura benchmarks, demonstrating superior efficiency for these symmetric systems.
• For Cyclic9, the algorithm required 378 primes of size 29 bits, took 1.3 days on one processor (5GB RAM), and 8.5 hours on six processors (16GB RAM), with a basis of ~1.4GB and coefficients of ~1600 digits.
• The probabilistic algorithm is competitive with Maple and about 3–4 times slower than Magma on single-core, but may outperform Magma on multi-core systems due to better parallelization.
• Memory usage is approximately twice the size of the final basis on ℚ, often significantly less than other algorithms, making it highly memory-efficient.
• The learning technique reduces redundant critical pair computation, improving performance especially on symmetric systems like Cyclic and Katsura, where many pairs reduce to zero.
• The implementation is only ~10K lines of code, significantly smaller than highly optimized systems like Fgb (200K lines), suggesting high maintainability and ease of integration.