Related papers: An effective estimate on Betti numbers

In this paper we generalize the approximation theorem for L^2-Betti numbers to an approximation theorem for center-valued Betti-numbers.

In this paper, we establish a new estimate (including lower and upper bounds) for an important quantity involved in the convergence analysis of smoothed aggregation algebraic multigrid methods. The new upper bound improves the existing…

The paper is devoted to systematic study of the $\chi$-capacity (underlying the classical capacity) of infinite dimensional quantum channels. An essential feature of this case is the natural appearance of the input constraints and infinite,…

We prove resolvent estimates in Schatten spaces for Laplace-Beltrami operators on compact manifolds at the critical exponent. Our proof only uses known bounds for the Hadamard parametrix.

An equivalent definition of entropic Ricci curvature on discrete spaces was given in terms of the global gradient estimate. With a particular choice of the density function $\rho$, we obtain a localized gradient estimate, which in turns…

In this paper we prove explicit estimates for the size of small lifts of points in homogeneous spaces. Our estimates are polynomially effective in the volume of the space and the injectivity radius.

The numerical analysis of gradient inclusions in a compact subset of $2\times 2$ diagonal matrices is studied. Assuming that the boundary conditions are reached after a finite number of laminations and using piecewise linear finite…

We use purely combinatorial arguments to give a formula to compute all graded Betti numbers of path ideals of line graphs and cycles. As a consequence we can give new and short proofs for the known formulas of regularity and projective…

Let $I_{n,m} = (x_1\cdots x_{m},x_2 \cdots x_{m+1},\ldots,x_{n+1}x_{n+2}\cdots x_{n+m})$ be the $m$-path ideal of a path of length $n + m-1$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{n+m}]$. We compute all the graded Betti…

We revisit the definition of effective local compactness, and propose an approach that works for arbitrary countably-based spaces extending the previous work on computable metric spaces. We use this to show that effective local compactness…

The Bethe approximation, or loopy belief propagation algorithm is a successful method for approximating partition functions of probabilistic models associated with a graph. Chertkov and Chernyak derived an interesting formula called Loop…

We prove a tight lower bound on the Betti numbers of tree and forest ideals and a tight upper bound on certain graded Betti numbers of squarefree monomial ideals.

On a real analytic Riemannian manifold a Grauert tube is an uniquely adapted complex structure defined on the tangent bundle. It is called entire if it may be defined on the whole tangent bundle. Here, we show that the geodesic flow of an…

We define and count lattice points in the moduli space of stable genus g curves with n labeled points. This extends a construction of the second author for the uncompactified moduli space. The enumeration produces polynomials with top…

We prove that any complete (and possibly non-compact) Riemannian manifold $M$ possesses infinitely many closed geodesics provided its free loop space has unbounded Betti numbers in degrees larger than the dimension of $M$, and there are no…

This survey paper was primarily written as as the support for a course pesented at the JNCF2025: it aims to present some material that illustrates the kind of estimates one can obtain in effective algebraic geometry, for affine polynomial…

Let (X,d) be a finite metric space. This paper first discusses the spectrum of the p-distance matrix of a finite metric space of p-negative type and then gives upper and lower bounds for the so called gap of a finite metric space of strict…

Let $M$ be a simply-connected $m$ dimensional manifold of finite type and $k$ a positif integer. In this paper we show that the rational Betti numbers of each component of the space of immersions of $M$ in $\mathbb{R}^{m+k}$, have…

Upper bounds on the topological Betti numbers of Vietoris-Rips complexes are established, and examples of such complexes with high Betti numbers are given.

A divide-and-conquer algorithm for computing the Betti numbers of finite $T_0$-spaces is presented. It extensively uses the Mayer-Vietoris sequence for open coverings. In the end, the computational costs for a parallelisation of this method…