[Paper Review] Equality of Graver bases and universal Gröbner bases of colored partition identities

Tristram Bogart, Raymond Hemmecke+1 more|arXiv (Cornell University)|Apr 6, 2010

Commutative Algebra and Its Applications12 references3 citations

TL;DR

This paper establishes that the universal Gröbner basis and Graver basis coincide for toric ideals associated with rational normal scrolls and homogeneous primitive colored partition identities, specifically for certain families of matrices including $S(n_1-1,\dots,n_c-1)$ and $H(n_1,\dots,n_c)$, under dominance conditions. The key contribution is a complete classification of equality in these families using combinatorial dominance and lifting arguments.

ABSTRACT

Associated to any vector configuration A is a toric ideal encoded by vectors in the kernel of A. Each toric ideal has two special generating sets: the universal Gröbner basis and the Graver basis. While the former is generally a proper subset of the latter, there are cases for which the two sets coincide. The most prominent examples among them are toric ideals of unimodular matrices. Equality of universal Gröbner basis and Graver basis is a combinatorial property of the toric ideal (or, of the defining matrix), providing interesting information about ideals of higher Lawrence liftings of a matrix. Nonetheless, a general classification of all matrices for which both sets agree is far from known. We contribute to this task by identifying all cases with equality within two families of matrices; namely, those defining rational normal scrolls and those encoding homogeneous primitive colored partition identities.

Motivation & Objective

• To classify all matrices for which the universal Gröbner basis and Graver basis of their associated toric ideals coincide.
• To extend the known class of matrices with equal universal Gröbner and Graver bases beyond unimodular matrices.
• To analyze the equality condition in the context of toric ideals from rational normal scrolls and homogeneous primitive colored partition identities.
• To provide a combinatorial characterization of when the two bases coincide using dominance and type bounds.
• To prove that equality holds for specific families of matrices, including $S(5,3,1,\dots,1)$, $S(5,2,\dots,2)$, $S(4,4,1,\dots,1)$, $H(6)$, and $H(5,2,\dots,2)$, under specified conditions.

Proposed method

• Use of the Graver basis $\mathcal{G}(A)$ as the set of all binomials in $I_A$ that are not reducible by any other binomial in $I_A$.
• Definition of the universal Gröbner basis $\mathrm{UGB}(A)$ as the union of all reduced Gröbner bases of $I_A$.
• Application of the dominance order on matrices to determine when $\mathrm{UGB}(A) = \mathcal{G}(A)$, particularly for scroll and colored partition identity matrices.
• Leveraging computational verification for small cases (e.g., $H(5,2,\dots,2)$ with 12 components) to establish base cases.
• Use of lifting and projection arguments via Corollary 4 to extend results from smaller to larger matrices by preserving Graver basis elements under column deletion and zero-row removal.
• Bounding the type of Graver basis elements (at most $4\cdot 5 - 7 = 13$) to ensure that elements in larger matrices can be projected to known equality cases.

Experimental results

Research questions

• RQ1 For which families of matrices does the universal Gröbner basis coincide with the Graver basis?
• RQ2 What combinatorial conditions on the matrix structure (e.g., dominance, type bounds) ensure equality of the two bases?
• RQ3 Can the equality result for small cases be extended to infinite families of matrices?
• RQ4 How do the toric ideals of rational normal scrolls and homogeneous primitive colored partition identities behave under dominance and lifting?
• RQ5 What is the role of the Lawrence lifting in connecting the equality of bases to higher-dimensional ideals?

Key findings

• The universal Gröbner basis equals the Graver basis for all matrices in the family $S(5,3,1,\dots,1)$, $S(5,2,\dots,2)$, $S(4,4,1,\dots,1)$, $H(6)$, and $H(5,2,\dots,2)$, regardless of the number of $1$s or $2$s.
• Equality holds for $S(n_1-1,\dots,n_c-1)$ if $n_1 \leq 4$, or if $n_1 = 5$ or $6$ and $n_2 \leq 3$, or if $n_1 = 5$ or $6$, $n_2 = 4$, and $n_3 \leq 2$, provided the matrix does not dominate $S(6)$, $S(5,4)$, or $S(4,3,2)$.
• For $H(n_1,\dots,n_c)$, equality holds if $n_1 \leq 3$, or $n_1 = 4$ or $5$ with $n_2 \leq 2$, or $n_1 = 6$ with $n_2 \leq 1$, provided the matrix does not dominate $H(7)$, $H(6,2)$, or $H(4,3)$.
• The result for $H(5,2,\dots,2)$ with $k > 12$ components is established by lifting Graver basis elements from $H(5,5,\dots,5)$ and projecting to the $k=12$ case, where equality is computationally verified.
• The type of any Graver basis element of $A_{H(5,5,\dots,5)}$ is bounded by $13$, enabling projection to smaller matrices without loss of generality.
• The equality of $\mathrm{UGB}(A)$ and $\mathcal{G}(A)$ is preserved under column deletion and zero-row removal, allowing inductive extension from finite to infinite families.