[Paper Review] Upper bounds for Extremal Betti Numbers of $t$-Spread Strongly Stable Ideals

Luca Amata, Antonino Ficarra+1 more|arXiv (Cornell University)|Feb 15, 2021

Commutative Algebra and Its Applications14 references4 citations

TL;DR

This paper determines the maximal number of extremal Betti numbers for $t$-spread strongly stable ideals in the polynomial ring $S = K[x_1, \dots, x_n]$, generalizing prior results for $t=1,2$. By constructing explicit families of $t$-spread monomials and analyzing their corner structures via combinatorial constraints, the authors establish tight upper bounds that depend on $n$, $t$, and the initial degree, unifying and extending earlier findings for squarefree and 2-spread cases.

ABSTRACT

We study the extremal Betti numbers of the class of $t$--spread strongly stable ideals. More precisely, we determine the maximal number of admissible extremal Betti numbers for such ideals, and thereby we generalize the known results for $t\in \{1,2\}$.

Motivation & Objective

• To determine the maximal number of extremal Betti numbers for $t$-spread strongly stable ideals in $S = K[x_1, \dots, x_n]$ for $t \geq 2$ and initial degree $\geq 2$.
• To generalize prior results on extremal Betti numbers for $t=1$ (squarefree) and $t=2$ (2-spread) strongly stable ideals.
• To characterize the positions and existence conditions of corners in $t$-spread strongly stable ideals using combinatorial constraints on monomial generation.
• To provide a unified framework for bounding extremal Betti numbers across all $t \geq 2$ by decomposing $n$ with respect to $t$.

Proposed method

• Constructing a family of $t$-spread monomials $\omega_0, \omega_1, \dots, \omega_{j_{\max}+1+\nu_{\max}}$ using recursive shifts in variable indices to generate $t$-spread strongly stable ideals.
• Defining corner monomials via the condition $k_j + t(\ell_j - 1) + 1 = n$ to ensure extremal Betti numbers occur at specified positions in the Betti diagram.
• Using the formula $j_{\max} = \left\lfloor \frac{n - (\ell_1 - 2)t}{t+1} \right\rfloor - 1$ to determine the number of initial corner monomials in the construction.
• Introducing a parameter $s = 2t - \left[ n - \left( (j_{\max}+1) + (\ell_1 - 2 + j_{\max})t \right) \right] $ to control the existence of the final monomial $\omega_{j_{\max}+1}$.
• Applying the characterization that a $t$-spread strongly stable ideal $I = B_t(\omega_0, \dots, \omega_m)$ has extremal Betti numbers exactly at the corners $\omega_j$, with $|\text{Corn}(I)| = k + \left\lfloor \frac{d-2}{t} \right\rfloor - (\ell_1 - 2)$.
• Verifying constructions with Macaulay2 packages and providing explicit examples for $n=138$, $t=11$, $\ell_1=5$.

Experimental results

Research questions

• RQ1 What is the maximal number of extremal Betti numbers possible for a $t$-spread strongly stable ideal of initial degree $\geq 2$ in $S = K[x_1, \dots, x_n]$ for $t \geq 2$?
• RQ2 How can the positions of extremal Betti numbers in the Betti diagram of such ideals be characterized combinatorially?
• RQ3 Under what conditions do $t$-spread monomials $\omega_j$ form a valid corner sequence in a $t$-spread strongly stable ideal?
• RQ4 Can the construction of extremal Betti number sequences be generalized beyond $t=1,2$ using a uniform framework?
• RQ5 What constraints on $n$, $t$, and initial degree $\ell_1$ ensure the existence of a $t$-spread strongly stable ideal with a specified number of extremal Betti numbers?

Key findings

• The maximal number of extremal Betti numbers for a $t$-spread strongly stable ideal of initial degree $\ell_1 \geq 2$ is $k + \left\lfloor \frac{d-2}{t} \right\rfloor - (\ell_1 - 2)$, where $n = d + kt$ with $1 \leq d \leq t$.
• For $t \geq 2$, the upper bound on extremal Betti numbers is achieved by constructing a $t$-spread strongly stable ideal $I = B_t(\omega_0, \dots, \omega_{j_{\max}+1+\nu_{\max}})$ with explicitly defined monomials $\omega_j$.
• The existence of the final monomial $\omega_{j_{\max}+1}$ depends on the parameter $s = 2t - \left[ n - \left( (j_{\max}+1) + (\ell_1 - 2 + j_{\max})t \right) \right] $, with $\omega_{j_{\max}+1}$ existing iff $j_{\max} + \ell_1 - 3 - s \geq \ell_1 - 2$.
• The construction generalizes known results: for $t=1$, the bound reduces to $k + \left\lfloor \frac{d-2}{1} \right\rfloor - (\ell_1 - 2)$, matching the result in [3]; for $t=2$, it matches the bound in [4].
• An explicit example with $n=138$, $t=11$, $\ell_1=5$ constructs an ideal with 9 extremal Betti numbers, confirming the bound via $j_{\max}=7$, $\nu_{\max}=0$, and $s=2$.
• Theorem 5.2 provides a necessary and sufficient condition for the existence of a $t$-spread strongly stable ideal with $r$ corners at specified positions $(k_j, \ell_j)$, requiring $k_j + t(\ell_j - 1) + 1 = n$ for all $j$.