Related papers: A sharper estimate on the Betti numbers of sets de…

This paper gives a sharp upper bound for the Betti numbers of a finitely generated multigraded $R$-module, where $R=\Bbbk [x_{1},...,x_{m}]$ is the polynomial ring over a field $\Bbbk$ in $m$ variables. The bound is given in terms of the…

Let $R=\mathbb{K}[x_1,\dots,x_n]$, a graded algebra $S=R/I$ satisfies $N_{k,p}$ if $I$ is generated in degree $k$, and the graded minimal resolution is linear the first $p$ steps, and the $k$-index of $S$ is the largest $p$ such that $S$…

The main goal of this paper is to size up the minimal graded free resolution of a homogeneous ideal in terms of its generating degrees. By and large, this is too ambitious an objective. As understood, sizing up means looking closely at the…

For any numerical semigroup $S$, there are infinitely many numerical symmetric semigroups $T$ such that $S=\frac{T}{2}$ is their half. We are studying the Betti numbers of the numerical semigroup ring $K[T]$ when $S$ is a 3-generated…

Suppose that $M$ is a finitely-generated graded module of codimension $c\geq 3$ over a polynomial ring and that the regularity of $M$ is at most $2a-2$ where $a\geq 2$ is the minimal degree of a first syzygy of $M$. Then we show that the…

We use stratified Morse theory for a manifold with corners to give a new bound for the sum of the Betti numbers of a hypersurface in R^n_> defined by a polynomial with n+l+1 terms.

We present a spectral sequence which efficiently computes Betti numbers of a closed semi-algebraic subset of RP^n defined by a system of quadratic inequalities and the image of the homology homomorphism induced by the inclusion of this…

Semi-algebraic set is a subset of the real space defined by polynomial equations and inequalities. In this paper, we consider the problem of deciding whether two given points in a semi-algebraic set are connected. We restrict to the case…

We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…

We prove explicit bounds on the radius of a ball centered at the origin which is guaranteed to contain all bounded connected components of a semi-algebraic set $S \subset \mathbbm{R}^k$ defined by a quantifier-free formula involving $s$…

Let $k_i\ (i=1,2,\ldots,t)$ be natural numbers with $k_1>k_2>\cdots>k_t>0$, $k_1\geq 2$ and $t<k_1.$ Given real numbers $\alpha_{ji}\ (1\leq j\leq t,\ 1\leq i\leq s)$, we consider polynomials of the shape…

We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why…

The following is shown : Let $S=\{a_1,a_2,..,a_{2n}\}$ be a subset of a totally ordered commutative semi-group $(G,*,\leq)$ with $a_1\leq a_2\leq...\leq a_{2n}$. Provided that a system of $n$ $a_{i_k} * a_{j_k}\ (a_{i_k}, a_{j_k} \in G ;\ 1…

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set is bounded from below (up to a multiplicative constant) by the logarithm of m-th Betti number (with respect to singular homology) of…

We asymptotically estimate from above the expected Betti numbers of random real hypersurfaces in smooth real projective manifolds. Our upper bounds grow as the square root of the degree of the hypersurfaces as the latter grows to infinity,…

Suppose X is any finite complex with vanishing L^2 Betti number. We prove upper bounds on the Betti numbers for regular coverings of X, sublinear in the order of covering. The bounds are sensitive to the Novikov-Shubin invariants of X, and…

$\newcommand{\Arr}{\mathcal{A}} \newcommand{\numS}{k} \newcommand{\ArrX}[1]{\Arr(#1)} \newcommand{\eps}{\varepsilon} \newcommand{\opt}{\mathsf{o}}$ For point sets $P_1, \ldots, P_\numS$, a set of lines $L$ is halving if any face of the…

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees…

Given a finite point set $P \subset \mathbb{R}^d$, a $k$-ary semi-algebraic relation $E$ on $P$ is the set of $k$-tuples of points in $P$, which is determined by a finite number of polynomial equations and inequalities in $kd$ real…

This article proves the existence of a bound on the sum of local Betti numbers of a real analytic germ by a polynomial function of its multiplicity. This result can be interpreted as a localization of the classical Oleinik-Petrovsky bound…