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Axioms for a theory of signature bases

Pierre Lairez

TL;DR

This work reframes Gröbner bases with signatures through an axiomatic lens to decouple theory from algorithms and extend applicability to broader settings such as modules, noncommutative rings, and non-Noetherian contexts. It introduces monomial spaces and monomial modules, together with prebases, signatures, and rewrite bases, to establish general notions of correctness and termination. A key result shows that a sigpoly set forming a module-based Gröbner basis is automatically a prebasis, enabling reuse of signatures across computations, while sigtrees provide a universal termination argument across various reduction strategies, including F4-style reductions. The framework applies to diverse contexts, including submodules and differential or noncommutative algebras, and shifts emphasis from heavy arithmetic checks to combinatorial, monomial-based criteria, facilitating broader applicability and robust algorithm design.

Abstract

Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, and giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.).

Axioms for a theory of signature bases

TL;DR

This work reframes Gröbner bases with signatures through an axiomatic lens to decouple theory from algorithms and extend applicability to broader settings such as modules, noncommutative rings, and non-Noetherian contexts. It introduces monomial spaces and monomial modules, together with prebases, signatures, and rewrite bases, to establish general notions of correctness and termination. A key result shows that a sigpoly set forming a module-based Gröbner basis is automatically a prebasis, enabling reuse of signatures across computations, while sigtrees provide a universal termination argument across various reduction strategies, including F4-style reductions. The framework applies to diverse contexts, including submodules and differential or noncommutative algebras, and shifts emphasis from heavy arithmetic checks to combinatorial, monomial-based criteria, facilitating broader applicability and robust algorithm design.

Abstract

Twenty years after the discovery of the F5 algorithm, Gröbner bases with signatures are still challenging to understand and to adapt to different settings. This contrasts with Buchberger's algorithm, which we can bend in many directions keeping correctness and termination obvious. I propose an axiomatic approach to Gröbner bases with signatures with the purpose of uncoupling the theory and the algorithms, and giving general results applicable in many different settings (e.g. Gröbner for submodules, F4-style reduction, noncommutative rings, non-Noetherian settings, etc.).
Paper Structure (12 sections, 6 theorems, 8 equations)

This paper contains 12 sections, 6 theorems, 8 equations.

Table of Contents

  1. Introduction
  2. Contribution
  3. Plan
  4. Acknowledgment
  5. Gröbner bases
  6. Rewriting systems
  7. Top reduction
  8. Monomial modules
  9. Gröbner bases
  10. Signatures
  11. Signature bases
  12. Prebases

Key Result

Theorem 6

Let $E$ be a subset of $M$. The following assertions are equivalent: (Characterization by leading monomials) (Characterization by rewriting) (Characterizations by confluence properties) (Characterizations by S-pairs)

Theorems & Definitions (25)

  • Definition 1: Monomial space, leading monomial, $\equiv_{\mathrm{lt}}$
  • Remark 2: Equivalent monomial spaces
  • Example 3
  • Definition 4: Top reduction, $\to_E$
  • Definition 5: Tail equivalence, $\smile_E$, $\mathrel{{\breve\uparrow}}_E$, $\mathrel{{\breve\downarrow}}_E$
  • Theorem 6: Buchberger's criterion for monomial spaces
  • Definition 7: Pivot basis
  • Lemma 8
  • Definition 9: Monomial module
  • Definition 10: Action on the set of monomials, divisibility
  • ...and 15 more