[Paper Review] A new conception for computing gröbner basis and its applications

Lei Huang|arXiv (Cornell University)|Dec 24, 2010

Polynomial and algebraic computation7 references20 citations

TL;DR

This paper introduces a unified TRB (Top Reductional Basis) algorithm framework that generalizes F5, extended F5, and GVW algorithms, enabling systematic analysis of their finite termination. It proposes the Mpair Criterion, which outperforms existing criteria in blocking useless S-polynomials and ensures all TRB pairs are computed correctly.

ABSTRACT

This paper presents a conception for computing gröbner basis. We convert some of gröbner-computing algorithms, e.g., F5, extended F5 and GWV algorithms into a special type of algorithm. The new algorithm's finite termination problem can be described by equivalent conditions, so all the above algorithms can be determined when they terminate finitely. At last, a new criterion is presented. It is an improvement for the Rewritten and Signature Criterion.

Motivation & Objective

• To unify diverse Gröbner basis algorithms (F5, extended F5, GVW) under a common computational framework.
• To resolve the long-standing open problem of finite termination for GVW and F5 algorithms by providing equivalent termination conditions.
• To develop a new filtering criterion (Mpair) that blocks more useless S-polynomials than existing Rewritten and Signature criteria.
• To establish a general platform for algorithm comparison, implementation, and future extension in Gröbner basis computation.

Proposed method

• Formalize a general TRB (Top Reductional Basis) algorithm that subsumes F5, extended F5, and GVW as special cases.
• Define key structures: pairs (u,v) ∈ P^d × P, signatures (lm(u)), and orders (monomial, signature, pair order).
• Introduce the Mpair Criterion: a pair [m,p] is blocked if it is neither initial nor an M-pair of the DONE set.
• Define M-pairs as minimal multiplied pairs [m,p] with m ≠ 1 and p not equivalent or less favorable than other pairs in signature and pair order.
• Use top reductions and a CheckStore mechanism that always accepts valid J-pairs passing MJCriterions (GSyzygy + Mpair).
• Prove that TRB-MJ algorithm computes all TRB pairs and only stores syzygy or TRB signatures, ensuring correctness and completeness.

Experimental results

Research questions

• RQ1 Under what conditions do F5, extended F5, and GVW algorithms terminate finitely?
• RQ2 Can a unified framework be constructed to analyze and compare the correctness and termination of modern Gröbner basis algorithms?
• RQ3 How does the Mpair Criterion compare to existing criteria (Rewritten, Signature) in blocking useless S-polynomials?
• RQ4 Can the Mpair Criterion detect non-TRB signatures that are missed by the Rewritten and Signature Criterions?
• RQ5 Is the TRB-MJ algorithm sufficient to compute all TRB pairs while avoiding redundant computations?

Key findings

• F5 and extended F5 algorithms always terminate finitely, as they are regular TRB algorithms.
• The GVW algorithm terminates finitely if the monomial order and signature order are almost compatible.
• The Mpair Criterion blocks more unnecessary pairs than the Rewritten and Signature Criterions, and can detect non-TRB signatures that those criteria miss.
• The Mpair Criterion ensures that only TRB signatures (and syzygy signatures) are computed, reducing the search space.
• All TRB pairs are computed by the TRB-MJ algorithm, as proven by the fact that every TRB signature is realized as an MJ-signature.
• The TRB-MJ algorithm is a regular TRB algorithm, and its CheckStore always returns true for valid J-pairs passing the MJCriterions.