[Paper Review] Simple signature-based Groebner basis algorithm

Vasily Galkin|arXiv (Cornell University)|May 28, 2012

Polynomial and algebraic computation2 references3 citations

TL;DR

This paper presents a simple signature-based Groebner basis algorithm that computes a Gröbner basis for an ideal by extending an existing basis with a new polynomial using labeled polynomials and a single signature-based discarding criterion. The method ensures correctness through a well-ordered signature comparison, achieves efficiency comparable to advanced algorithms like F5 and TRB-MJ, and enables straightforward implementation without relying on S-polynomials or complex data structures.

ABSTRACT

This paper presents an algorithm for computing Groebner bases based upon labeled polynomials and ideas from the algorithm F5. The main highlights of this algorithm compared with analogues are simplicity both of the algorithm and of the its correctness proof achieved without loss of the efficiency. This leads to simple implementation which performance is in par with more complex analogues

Motivation & Objective

• To develop a signature-based Groebner basis algorithm that is simpler to implement and verify than existing methods such as F5.
• To maintain high computational efficiency comparable to advanced signature-based algorithms like TRB-MJ and SB while reducing algorithmic complexity.
• To eliminate reliance on S-polynomials in the correctness proof, enabling a clearer algebraic interpretation of signature-based methods.
• To simplify the implementation process by minimizing data structures and using a single ordering for both discarding criteria and reductor selection.
• To demonstrate that the algorithm produces a minimal S-basis and a correct Gröbner basis with performance on par with state-of-the-art incremental algorithms.

Proposed method

• The algorithm uses labeled polynomials of the form $ h = ( ho, p) $, where $ ho $ is the signature and $ p $ is the polynomial, with the condition that $ ho $ corresponds to a multiple of the new generator $ f $ modulo the ideal $ I_0 $.
• It introduces a partial order $ <_{ ext{H}} $ on labeled polynomials defined by $ ( ho_1, p_1) <_{ ext{H}} ( ho_2, p_2) $ iff $ ext{HM}(p_1) ho_2 riangleq ext{HM}(p_2) ho_1 $, which guides both reduction and discarding decisions.
• Signature-safe reduction is performed only when the reductor’s signature is strictly smaller in $ <_{ ext{H}} $, ensuring that reductions do not compromise the signature structure.
• A single discarding criterion in step 4d uses the $ <_{ ext{H}} $ order to avoid unnecessary computations, replacing multiple criteria used in F5 and its variants.
• The algorithm maintains an invariant that all polynomials in the result set $ R $ are minimal S-basis elements, and correctness is proven via signature representations and invariants.
• The implementation uses a single reductor selection strategy based on the $ <_{ ext{H}} $ order, simplifying the reduction process and enabling efficient integration into computer algebra systems.

Experimental results

Research questions

• RQ1 Can a signature-based Groebner basis algorithm be designed with a single, unified discarding criterion that maintains efficiency comparable to multi-criteria algorithms like TRB-MJ?
• RQ2 Is it possible to prove correctness of a signature-based algorithm without relying on S-polynomials, thereby simplifying the theoretical framework?
• RQ3 To what extent can algorithmic simplicity be achieved without sacrificing computational performance in signature-based Groebner basis computation?
• RQ4 How does the use of a single ordering for both reduction and discarding affect implementation complexity and integration into existing computer algebra systems?
• RQ5 Does the proposed algorithm produce a minimal S-basis and a correct Gröbner basis with performance matching or exceeding that of established incremental signature-based algorithms?

Key findings

• The algorithm correctly computes a Gröbner basis for the ideal $ I = (g_1, reak ext{...}, g_m, f) $, as proven by the final invariant that $ R $ is an S-basis.
• The number of polynomials in the resulting Gröbner basis is not greater than that produced by other incremental signature-based algorithms, indicating minimal output size.
• The execution time of the SimpleSignatureGroebner algorithm is not greater than that of other signature-based incremental algorithms, including F5 and TRB-MJ, as confirmed by empirical comparisons.
• The correctness proof is formulated without using S-polynomials, relying instead on signature representations and the $ <_{ ext{H}} $ order, which allows for a clearer algebraic interpretation.
• The algorithm’s implementation in C++ using Singular 3-1-4 took only 8 hours to write from scratch, significantly less than time spent on implementing other F5-like algorithms.
• The algorithm’s simplicity enables direct integration into computer algebra systems due to minimal data structures and a single, consistent ordering for both reduction and discarding.