[Paper Review] A Groebner basis for the secant ideal of the second hypersimplex

Seth Sullivant|ArXiv.org|Apr 17, 2008

Commutative Algebra and Its Applications9 references3 citations

TL;DR

This paper computes a Gröbner basis for the secant ideal of the toric ideal associated with the second hypersimplex Δ(2,n) under any circular term order, proving that the initial ideal requires polynomials of odd degree up to n. This confirms the 2-delightful property of the circular term order, resolving a conjecture by Drton, Sturmfels, and Sullivant, and further establishes a Gröbner basis for the symbolic square of the ideal using master polynomials and off-diagonal minors.

ABSTRACT

We determine a Groebner basis for the secant ideal of the toric ideal associated to the second hypersimplex, with respect to any circular term order. The Groebner basis of the secant ideal requires polynomials of odd degree up to n. This shows that the circular term order is 2-delightful, resolving a conjecture of Drton, Sturmfels, and the author. The proof uses Groebner degenerations for secant ideals, combinatorial characterizations of the secant ideals of monomial ideals, and the relations between secant ideals and prolongations.

Motivation & Objective

• To compute a Gröbner basis for the secant ideal $I_{n}^{ ext{\{2\}}}$ of the toric ideal associated with the second hypersimplex $\Delta(2,n)$.
• To resolve the conjecture that the circular term order is 2-delightful by showing that the initial ideal of the secant ideal requires polynomials of odd degree up to $n$.
• To establish a Gröbner basis for the symbolic square $I_{n}^{(2)}$ using connections between secant ideals and prolongations.
• To apply the 'delightful' strategy via Gröbner degenerations and combinatorial characterizations of secant ideals of monomial ideals.

Proposed method

• Employing the 'delightful' strategy from Sturmfels and Sullivant, which links the initial ideal of the secant ideal to the secant ideal of the initial ideal.
• Using Gröbner degenerations to relate $I_{n}^{ ext{\{2\}}}$ to the initial ideal ${\rm in}_{\prec}(I_n)^{\text{\{2\}}}$ via the inclusion ${\rm in}_{\prec}(I_n^{ ext{\{2\}}}) \subseteq {\rm in}_{\prec}(I_n)^{\text{\{2\}}}$.
• Defining master polynomials $f_{\mathbf{i},\mathbf{j}}$ associated with admissible sequences of odd length $\geq 5$, whose leading terms are cycle monomials in the noncrossing graph of $K_n$.
• Showing that the leading term of each master polynomial is the cycle monomial $\prod_{l=1}^{2k+1} x_{i_l j_{l+k-1}}$, using Gröbner reduction with respect to the circular Gröbner basis of $I_n$.
• Establishing that 3×3 off-diagonal minors in the matrix of variables $x_{ij}$ have initial terms corresponding to monomials $x_{i_1 i_6}x_{i_2 i_5}x_{i_3 i_4}$, which are the initial terms of degree-3 generators.
• Using the containment ${\rm in}_{\prec}(I_n)^{(2)} = {\rm in}_{\prec}(I_n)^2 + {\rm in}_{\prec}(I_n)^{\text{\{2\}}}$ to deduce a Gröbner basis for $I_n^{(2)}$.

Experimental results

Research questions

• RQ1 Does the circular term order yield a 2-delightful Gröbner basis for the secant ideal $I_n^{\text{\{2\}}}$ of the second hypersimplex?
• RQ2 What is the minimal degree of polynomials required in a Gröbner basis for $I_n^{\text{\{2\}}}$ under a circular term order?
• RQ3 Can the symbolic square $I_n^{(2)}$ be described via a Gröbner basis using the same term order?
• RQ4 How do master polynomials and off-diagonal minors contribute to generating the initial ideal of $I_n^{\text{\{2\}}}$?
• RQ5 What is the combinatorial structure of the initial ideal ${\rm in}_{\prec}(I_n)^{\text{\{2\}}}$ in terms of noncrossing and crossing edges in the circular embedding of $K_n$?

Key findings

• The secant ideal $I_n^{\text{\{2\}}}$ admits a Gröbner basis under any circular term order, with generators including 3×3 off-diagonal minors, degree-3 master polynomials, and products of pairs of 2×2 minors.
• The initial ideal ${\rm in}_{\prec}(I_n^{\text{\{2\}}})$ requires polynomials of odd degree up to $n$, confirming that the circular term order is 2-delightful.
• Each degree-3 monomial of the form $x_{i_1 i_6}x_{i_2 i_5}x_{i_3 i_4}$ arises as the initial term of a 3×3 off-diagonal minor in the matrix of variables $x_{ij}$.
• The master polynomial $f_{\mathbf{i},\mathbf{j}}$ associated with an admissible odd sequence of length $2k+1 \geq 5$ has leading term equal to the cycle monomial $\prod_{l=1}^{2k+1} x_{i_l j_{l+k-1}}$.
• The symbolic square $I_n^{(2)}$ satisfies $I_n^{(2)} = I_n^2 + I_n^{\text{\{2\}}}$, and its Gröbner basis consists of the same generators as $I_n^{\text{\{2\}}}$, plus products of 2×2 minors.
• The initial ideal ${\rm in}_{\prec}(I_n)^{(2)}$ is generated by degree-3 cycles in the noncrossing graph and products of noncrossing edge pairs, matching the initial terms of the generators of $I_n^{(2)}$.