A squarefree monomial ideal is called an $f$-ideal if its Stanley–Reisner and facet simplicial complexes have the same $f$-vector. We show that $f$-ideals generated in a fixed degree asymptotic density zero when number of variables goes to infinity. also provide novel algorithms construct small degrees.

Let S = K[x1, . . . , xn] denote the polynomial ring in n variables over a field K with each deg xi = 1. Let ∆ be a simplicial complex on [n] = {1, . . . , n} and I∆ ⊂ S its Stanley–Reisner ideal. We write ∆e for the exterior algebraic shifted complex of ∆ and ∆c for a combinatorial shifted complex of ∆. Let βii+j(I∆) = dimK Tori(K, I∆)i+j denote the graded Betti numbers of I∆. In the present p...

Let C ⊂ N be an affine semigroup, and R = K[C] its semigroup ring. This paper is a collection of various results on “C-graded” R-modules M = ⊕ c∈C Mc, especially, monomial ideals of R. For example, we show the following: If R is normal and I ⊂ R is a radicalmonomial ideal (i.e., R/I is a generalization of Stanley-Reisner rings), then the sequentially Cohen-Macaulay property of R/I is a topologi...

Given a simplicial hyperplane arrangement H and a subspace arrangement A embedded in H, we define a simplicial complex ∆A,H as the subdivision of the link of A induced by H. In particular, this generalizes Steingŕımsson’s coloring complex of a graph. We do the following: (1) When A is a hyperplane arrangement, ∆A,H is shown to be shellable. As a special case, we answer affirmatively a question ...

Let M be a monomial ideal in the polynomial ring S = k[x1, . . . , xn] over a field k. We are interested in the problem of resolving S/M over S. The difficulty in resolving minimally is reflected in the fact that the homology of arbitrary simplicial complexes can be encoded (via the Stanley-Reisner correspondence) into the multigraded Betti numbers of S/M , [St]. In particular, the minimal free...

Associated to a simple undirected graph G is a simplicial complex ∆G whose faces correspond to the independent sets of G. We call a graph G shellable if ∆G is a shellable simplicial complex in the non-pure sense of Björner-Wachs. We are then interested in determining what families of graphs have the property that G is shellable. We show that all chordal graphs are shellable. Furthermore, we cla...

In computational complexity, a complexity class is given by a set of problems or functions, and a basic challenge is to show separations of complexity classes A 6= B especially when A is known to be a subset of B. In this paper we introduce a homological theory of functions that can be used to establish complexity separations, while also providing other interesting consequences. We propose to a...