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Reduced Groebner Bases With Double Exponential Cardinality

Archana S Morye, Sreenanda S B, Prakash Saivasan

TL;DR

The paper investigates the potential size of reduced Gröbner bases under various monomial orderings, proving that there exist families of polynomials with only $O(n)$ generators whose reduced Gröbner bases can be as large as $e(n)=2^{2^n}$. It constructs a Meyers–Meyer-inspired framework $\mathcal{F}(n)$ and a two-condition order-theoretic criterion on monomial orders that guarantees a double-exponential lower bound on $|\texttt{Gb}_{\preccurlyeq}(\mathcal{F}(n))|$, including high-degree leading terms. It then demonstrates that standard orderings—Lex, Degree Lexicographic, and Weighted—satisfy this criterion, establishing double-exponential growth for these common choices. The results highlight inherent worst-case complexity in computing Gröbner bases over $\mathbb{F}_2$ and provide a structured way to induce large bases from scalable polynomial families.

Abstract

In this article, we investigate the cardinality of Groebner bases under various monomial orderings. We identify a family of polynomials F and a criterion such that the reduced Groebner basis of F is double exponential in cardinality with respect to any monomial ordering which satisfies this criterion. We also show that the said criterion is satisfied by orderings such as the lexicographic, degree lexicographic and weighted orderings.

Reduced Groebner Bases With Double Exponential Cardinality

TL;DR

The paper investigates the potential size of reduced Gröbner bases under various monomial orderings, proving that there exist families of polynomials with only generators whose reduced Gröbner bases can be as large as . It constructs a Meyers–Meyer-inspired framework and a two-condition order-theoretic criterion on monomial orders that guarantees a double-exponential lower bound on , including high-degree leading terms. It then demonstrates that standard orderings—Lex, Degree Lexicographic, and Weighted—satisfy this criterion, establishing double-exponential growth for these common choices. The results highlight inherent worst-case complexity in computing Gröbner bases over and provide a structured way to induce large bases from scalable polynomial families.

Abstract

In this article, we investigate the cardinality of Groebner bases under various monomial orderings. We identify a family of polynomials F and a criterion such that the reduced Groebner basis of F is double exponential in cardinality with respect to any monomial ordering which satisfies this criterion. We also show that the said criterion is satisfied by orderings such as the lexicographic, degree lexicographic and weighted orderings.

Paper Structure

This paper contains 16 sections, 20 theorems, 32 equations, 3 figures.

Table of Contents

  1. Preliminaries
  2. Characterising the highest terms of a Gröbner basis
  3. A criterion to obtain double exponential Gröbner bases
  4. Base case
  5. Case $k > 1$
  6. Large cardinality for reduced Gröbner basis
  7. Lexicographic ordering
  8. Degree lexicographic ordering
  9. Weighted ordering
  10. Missing Proofs
  11. Proof of Proposition \ref{['prop:mayr1']}
  12. Case $m = 0$
  13. Case $m +1$
  14. Proof of Lemma \ref{['lem:mayrmeyer']}
  15. Case $i=0$
  16. ...and 1 more sections

Key Result

Lemma 1

Let $\mathcal{R}(F)$ be the set defined as above. If $g\in \langle F \rangle$, then $\mathcal{HT}(g) \in\mathcal{R}(F)$. Furthermore, $\mathcal{R}(F)$ is upward closed with respect to the order $\triangleleft$.

Figures (3)

  • Figure 1: Illustration for Theorem \ref{['thm:main']}
  • Figure 2: $s + \ell c^{m_1} \bar{c}^{m_2} \in \langle \mathcal{F} \rangle$
  • Figure 3: Summary of the proof of Proposition \ref{['prop:mayr1']} and Lemma \ref{['lem:mayrmeyer']}

Theorems & Definitions (53)

  • Definition 1: Monomial Ordering $\preccurlyeq$
  • Definition 2: Monomial Ordering $\preccurlyeq$ for Polynomials
  • Example 1
  • Definition 3
  • Example 2
  • Definition 4: $\triangleleft$ ordering over $X^\oplus$
  • Definition 5: $\triangleleft$ ordering over $\mathbb{F}_2[X]$
  • Example 3
  • Definition 6
  • Example 4
  • ...and 43 more