Generic initial ideals and squeezed spheres

Balanced squeezed Complexes

Given any order ideal $U$ consisting of color-squarefree monomials involving variables with $d$ colors, we associate to it a balanced $(d-1)$-dimensional simplicial complex $Δ_{\mathrm{bal}}(U)$ that we call a balanced squeezed complex. In fact, these complexes have properties similar to squeezed balls as introduced by Kalai and the more general squeezed complexes, introduced by the authors. We show that any balanced squeezed complex is vertex-decomposable and that its flag $h$-vector can be read off from the underlying order ideal. Moreover, we describe explicitly its Stanley-Reisner ideal $I…

Resolutions of co-letterplace ideals and generalizations of Bier spheres

We give the resolutions of co-letterplace ideals of posets in a completely explicit, very simple form. This generalizes and simplifies a number of linear resolutions in the literature, among them the Eliahou-Kervaire resolutions of strongly stable ideals generated in a single degree. Our method is based on a general result of K. Yanagawa using the canonical module of a Cohen-Macaulay Stanley-Reisner ring. We discuss in detail how the canonical module may effectively be computed, and from this derive directly the resolutions. A surprising consequence is that we obtain a large class of simplicia…

Vanishing ideals of binary Hamming spheres

We consider the simplified Algebraic Normal Form (sANF) of Boolean functions vanishing on Hamming spheres centred at zero and the associated sANF vector. We show that this vector is periodic, leading to an efficient computation of the sANF and to specific formulas for particular cases. Moreover, we explicitly provide a connection to the binary M{ö}bius transform of the elementary symmetric functions. We conclude by presenting a method based on polynomial evaluation to bound the minimum distance of binary nonlinear codes. The same method can be used to compute the minimum distance and the weigh…

Algebraic shifting of strongly edge decomposable spheres

Recently, Nevo introduced the notion of strongly edge decomposable spheres. In this paper, we characterize the algebraic shifted complex of those spheres. Algebraically, this result yields the characterization of the generic initial ideal of the Stanley--Reisner ideal of Gorenstein* complexes having the strong Lefschetz property in characteristic 0.

作为独立复形的球面与球体

Spheres and balls as independence complexes

该研究探讨加须、嫁接构造的单纯复形的独立复形，证明其为广义Bier球（同胚于球或球面），并分类了极优覆盖图独立复形的同胚类型。

Artin单项理想极化研究

Polarizations of Artin monomial ideals

研究证明Artin单项理想极化定义了三角形球体，推翻了Almousa等人的猜想，并展示了极化与Cohen-Macaulay子复形的关系，以及极化与超立方体分类的联系。