Related papers: Betti numbers are testable

Let $K$ be a field and let $S=K[x_1,\dots,x_n]$ be a standard polynomial ring over a field $K$. We characterize the extremal Betti numbers, values as well positions, of a $t$-spread strongly stable ideal of $S$. Our approach is…

We prove an inequality between the sum of the Betti numbers of a complex projective manifold and its total curvature, and we characterize the complex projective manifolds whose total curvature is minimal. These results extend the classical…

Random simplicial complexes, as generalizations of random graphs, have become increasingly popular in the literature in recent years. In this paper, we consider a new model for a random simplicial complex that was introduced in…

We survey results related to the magnitude of the Betti numbers of numerical semigroup rings and of their tangent cones.

We construct certain $\theta$-series associated to number fields and prove that for number fields of degree less than equal to 4, these $\theta$-series are number field invariants. We also investigate whether or not the collection of…

Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone…

We give conjectures on the possible graded Betti numbers of Cohen-Macaulay modules up to multiplication by positive rational numbers. The idea is that the Betti diagrams should be non-negative linear combinations of pure diagrams. The…

We prove that the thin parts of arithmetically defined locally symmetric space take up a negligible part of their volume and deduce asymptotic results on their Betti numbers.

An explicit construction of locally testable codes of constant rate, constant distance and constant number of queries is given. Hence answering affirmatively the $c^3$-problem.

We show that if a group G is finitely presented and nilpotent-by-abelian-by-finite, then there is an upper bound on the first betti number of M as M runs through all subgroups of finite index in G.

We study the homological properties of $\Delta_{\mathbf{r}}(n_1, \dots, n_e)$, a simplicial complex formed by sequentially gluing complete graphs along $(r_i-1)$-simplices. This construction generates precisely the chordal clique complexes,…

For primes $p$, we investigate an $\mathbb{F}_p$-version of simplicial volume and compare these invariants with their siblings over other coefficient rings. We will also consider the associated gradient invariants, obtained by stabilisation…

Given any finite simplicial complex \Delta, we show how to construct a new simplicial complex \Delta_{\chi} that is balanced and vertex decomposable. Moreover, we show that the h-vector of the simplicial complex \Delta_{\chi} is precisely…

We compute the graded Betti numbers for the toric ideal of a family of graphs constructed by adjoining a cycle to a complete bipartite graph. The key observation is that this family admits an initial ideal which has linear quotients. As a…

Let $G$ be a cycle or a complete bipartite graph. We show that the binomial edge ideal $J_{G}$ and its initial ideal with respect to the lexicographic order have the same extremal Betti number.

We prove the boundedness of the time derivative in the parabolic Signorini problem, as well as establish its H\"older continuity at regular free boundary points.

We show that the local-global divisibility in commutative algebraic groups defined over number fields can be tested on sets of primes of arbitrary small density, i.e. stable and persistent sets. We also give a new description of the…

We propose a refinement of the Betti numbers and of the homology with coefficients in a field of a compact ANR in the presence of a continuous real valued function. The refinement of Betti numbers consists of finite configurations of points…

Let $S = K[x_1, ..., x_n]$ denote the polynomial ring in $n$ variables over a field $K$ with each $\deg x_i = 1$. Let $\Delta$ be a simplicial complex on $[n] = \{1, ..., n \}$ and $I_\Delta \subset S$ its Stanley--Reisner ideal. We write…

We show that birational smooth complex projective varieties with numerically effective canonical bundles along the exceptional loci have the same Betti numbers. In particular, birational smooth minimal models share the same Betti numbers.…