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An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences

Kosuke Sakata, Tsuyoshi Takagi

TL;DR

The work presents LGB, an efficient Hilbert-driven algorithm for extracting the leading monomials of a minimal Gröbner basis of a generic sequence, leveraging the Moreno–Socías conjecture that $\mathrm{LT}(I)$ is weakly reverse-lexicographic and a closed-form Hilbert-series expression. By constructing the leading-monomial set degree by degree and matching Hilbert functions, LGB avoids full polynomial reductions. The paper introduces multiple refinements—degree-based termination, tiered monomial filtering, and selective divisor checks—that significantly reduce search space, computation time, and memory usage. Comprehensive experiments demonstrate substantial performance gains over standard Gröbner-basis computations, particularly when the number of equations exceeds the number of variables. The approach provides a practical, scalable route to predict and assemble leading monomials of minimal Gröbner bases in generic settings, with broad implications for symbolic computation and algebraic geometry workflows.

Abstract

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties conjectured to hold for generic sequences-specifically, that their leading monomial ideals are weakly reverse lexicographic and that their Hilbert series follow a known closed-form expression. The algorithm incrementally constructs the set of leading monomials degree by degree by comparing Hilbert functions of monomial ideals with the expected Hilbert series of the input ideal. To enhance computational efficiency, we introduce several optimization techniques that progressively narrow the search space and reduce the number of divisibility checks required at each step. We also refine the loop termination condition using degree bounds, thereby avoiding unnecessary recomputation of Hilbert series. Experimental results confirm that the proposed method substantially reduces both computation time and memory usage compared to conventional Groebner basis computations for computing the leading monomials of a minimal Groebner basis of generic sequences.

An Algorithm for Computing the Leading Monomials of a Minimal Groebner Basis of Generic Sequences

TL;DR

The work presents LGB, an efficient Hilbert-driven algorithm for extracting the leading monomials of a minimal Gröbner basis of a generic sequence, leveraging the Moreno–Socías conjecture that is weakly reverse-lexicographic and a closed-form Hilbert-series expression. By constructing the leading-monomial set degree by degree and matching Hilbert functions, LGB avoids full polynomial reductions. The paper introduces multiple refinements—degree-based termination, tiered monomial filtering, and selective divisor checks—that significantly reduce search space, computation time, and memory usage. Comprehensive experiments demonstrate substantial performance gains over standard Gröbner-basis computations, particularly when the number of equations exceeds the number of variables. The approach provides a practical, scalable route to predict and assemble leading monomials of minimal Gröbner bases in generic settings, with broad implications for symbolic computation and algebraic geometry workflows.

Abstract

We present an efficient algorithm for computing the leading monomials of a minimal Groebner basis of a generic sequence of homogeneous polynomials. Our approach bypasses costly polynomial reductions by exploiting structural properties conjectured to hold for generic sequences-specifically, that their leading monomial ideals are weakly reverse lexicographic and that their Hilbert series follow a known closed-form expression. The algorithm incrementally constructs the set of leading monomials degree by degree by comparing Hilbert functions of monomial ideals with the expected Hilbert series of the input ideal. To enhance computational efficiency, we introduce several optimization techniques that progressively narrow the search space and reduce the number of divisibility checks required at each step. We also refine the loop termination condition using degree bounds, thereby avoiding unnecessary recomputation of Hilbert series. Experimental results confirm that the proposed method substantially reduces both computation time and memory usage compared to conventional Groebner basis computations for computing the leading monomials of a minimal Groebner basis of generic sequences.
Paper Structure (23 sections, 9 theorems, 18 equations, 2 figures, 6 tables, 3 algorithms)

This paper contains 23 sections, 9 theorems, 18 equations, 2 figures, 6 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Polynomial Ring
  4. Gröbner Bases
  5. Hilbert Series
  6. Computation of Hilbert Series
  7. Generic Sequences
  8. Our Algorithm for the Leading Monomials of a Minimal Gröbner Basis of a Generic Sequence
  9. Toy Example
  10. Improvements to LGB by Modifying the Loop Termination Condition
  11. Modification of the Loop Termination Condition When $n \leq m$
  12. Modification of the Loop Termination Condition When $n > m$
  13. Improvement of LGB by Reducing the Number of Monomials to be Checked for Divisibility by $\langle L_G \rangle$
  14. tier 1
  15. tier 2
  16. ...and 8 more sections

Key Result

proposition 1

Let $I \subset R$ be an ideal, and let $G \subset I$ be a finite set of polynomials. Then, for all $d \geq 0$, Moreover, $G$ is a Gröbner basis of $I$ if and only if equality holds for all $d \geq 0$.

Figures (2)

  • Figure 1: Comparison of timing between Gröbner basis computation (GB) and LGB
  • Figure 2: Comparison of memory consumption between Gröbner Basis computation (GB) and LGB

Theorems & Definitions (22)

  • definition 1
  • definition 2
  • definition 3: Hilbert function and Hilbert series
  • proposition 1: CLO
  • proposition 2: CLO
  • remark 1
  • definition 4: Weakly reverse-lexicographic ideal
  • definition 5
  • theorem 1: Pardue
  • theorem 2
  • ...and 12 more