Betti Numbers of Semialgebraic Sets Defined by Quantifier-Free Formulae

Let X be a semialgebraic set in Rⁿ defined by a Boolean combination of atomic formulae of the kind h * 0 where * \in { >, \ge, = }, deg(h) < d, and the number of distinct polynomials h is k. We prove that the sum of Betti numbers of X is less than O(k²d)ⁿ.     

Computing the Top Betti Numbers of Semialgebraic Sets Defined by Quadratic Inequalities in Polynomial Time

For any ℓ>0, we present an algorithm which takes as input a semi-algebraic set, S, defined by P ₁≤0,…,P _s ≤0, where each P _i ∈R[X ₁,…,X _k ] has degree≤2, and computes the top ℓ Betti numbers of S, b _k−1(S),…,b _k−ℓ (S), in polynomial time. The complexity of the algorithm, stated more precisely, is . For fixed ℓ, the complexity of the algorithm can be expressed as , which is polynomial in the input parameters s and k. To our knowledge this is the first polynomial time algorithm for computing nontrivial topological invariants of semialgebraic sets in R ^k defined by polynomial inequalities, where the number of inequalities is not fixed and the polynomials are allowed to have degree greater than one. For fixed s, we obtain, by letting ℓ=k, an algorithm for computing all the Betti numbers of S whose complexity is . An erratum to this article can be found at     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

Different Bounds on the Different Betti Numbers of Semi-Algebraic Sets

   Abstract. A classic result in real algebraic geometry due to Oleinik—Petrovskii, Thom and Milnor, bounds the topological complexity (the sum of the Betti numbers) of basic semi-algebraic sets. This bound is tight as one can construct examples having that many connected components. However, till now no significantly better bounds were known on the individual higher Betti numbers. We prove better bounds on the individual Betti numbers of basic semi-algebraic sets, as well as arrangements of algebraic hypersurfaces. As a corollary we obtain a polynomial bound on the highest Betti numbers of basic semi-algebraic sets defined by quadratic inequalities.     

Basicness of Semialgebraic Sets

This paper is concerned with the problem of deciding whether a semialgebraic set S of an algebraic variety X over R is basic. Furthermore, in such a case, we decide what is the sharp number of inequalities defining S. For that, it suffices to desingularize X, as well as the boundary of S, and then ask the same question for the trace of S on its boundary. In this way, after a finite number of blowing-ups, we lower the dimension of the data and by induction we get a finite decision procedure to solve this problem. Decidability of other known criteria is also analyzed.     

曲率与Betti数

本文中，我们首先根据经典的Ricci曲率与Betti数的S．Bochner定理得到了ε－极小Riemann浸入子流形的数量曲率与Betti数的结果。然后，我们考虑了紧致连通Riemann流形中曲率与Betti数之间的关系，推广 了经典的S．Bochner定理。     

On Bounding the Betti Numbers and Computing the Euler Characteristic of Semi-Algebraic Sets

In this paper we prove new bounds on the sum of the Betti numbers of closed semi-algebraic sets and also give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets. Given a closed semi-algebraic set S R ^k defined as the intersection of a real variety, Q=0, deg(Q)≤d, whose real dimension is k', with a set defined by a quantifier-free Boolean formula with no negations with atoms of the form P _i =0, P _i ≥ 0, P _i ≤ 0, deg(P _i ) ≤ d, 1≤ i≤ s, we prove that the sum of the Betti numbers of S is bounded by s ^k' (O(d)) ^k . This result generalizes the Oleinik—Petrovsky—Thom—Milnor bound in two directions. Firstly, our bound applies to arbitrary unions of basic closed semi-algebraic sets, not just for basic semi-algebraic sets. Secondly, the combinatorial part (the part depending on s ) in our bound, depends on the dimension of the variety rather than that of the ambient space. It also generalizes the result in [4] where a similar bound is proven for the number of connected components. We also prove that the sum of the Betti numbers of S is bounded by s ^k' 2 ^{O(k2 m4)} in case the total number of monomials occurring in the polynomials in is m. Using the tools developed for the above results, as well as some additional techniques, we give the first single exponential time algorithm for computing the Euler characteristic of arbitrary closed semi-algebraic sets. Received September 9, 1997, and in revised form March 18, 1998, and October 5, 1998.     

Betti Numbers of Hypergraphs

In this article, we study some algebraic properties of hypergraphs, in particular their Betti numbers. We define some different types of complete hypergraphs, which to the best of our knowledge are not previously considered in the literature. Also, in a natural way, we define a product on hypergraphs, which in a sense is dual to the join operation on simplicial complexes. For such product, we give a general formula for the Betti numbers, which specializes neatly in case of linear resolutions.     

A Sharper Estimate on the Betti Numbers of Sets Defined by Quadratic Inequalities

In this paper we consider the problem of bounding the Betti numbers, b _i (S), of a semi-algebraic set S⊂ℝ ^k defined by polynomial inequalities P ₁≥0,…,P _s ≥0, where P _i ∈ℝ[X ₁,…,X _k ], s<k, and deg (P _i )≤2, for 1≤i≤s. We prove that for 0≤i≤k−1,

Integer Optimization on Convex Semialgebraic Sets

Let Y be a convex set in Re ^k defined by polynomial inequalities and equations of degree at most d ≥ 2 with integer coefficients of binary length at most l . We show that if the set of optimal solutions of the integer programming problem min is not empty, then the problem has an optimal solution of binary length ld ^O(k4) . For fixed k , our bound implies a polynomial-time algorithm for computing an optimal integral solution y ^* . In particular, we extend Lenstra's theorem on the polynomial-time solvability of linear integer programming in fixed dimension to semidefinite integer programming. Received August 3, 1998, and in revised form March 22, 1999.     

Shifting Operations and Graded Betti Numbers

The behaviour of graded Betti numbers under exterior and symmetric algebraic shifting is studied. It is shown that the extremal Betti numbers are stable under these operations. Moreover, the possible sequences of super extremal Betti numbers for a graded ideal with given Hilbert function are characterized. Finally it is shown that over a field of characteristic 0, the graded Betti numbers of a squarefree monomial ideal are bounded by those of the corresponding squarefree lexsegment ideal.     

Curvature, Diameter and Bounded Betti Numbers

In this paper, we introduce the notion of bounded Betti numbers, and show that the bounded Betti numbers of a closed Riemannian n-manifold (M, g) with Ric (M) ≥ -(n - 1) and Diam (M) ≤ D are bounded by a number depending on D and n. We also show that there are only finitely many isometric isomorphism types of bounded cohomology groups (H*(M), || · ||∞) among closed Riemannian manifold (M, g) with K(M) ≥-1 and Diam (M) ≤ D.     

Betti Numbers of Path Ideals of Trees

Let R = k[x₁,…, x_n], where k is a field. The path ideal (of length t) of a directed graph G is the monomial ideal, denoted by I_t(G), whose generators correspond to the directed paths of length t in G. We determine all the graded Betti numbers of the path ideal of a directed rooted tree with respect to some graphical terms.     

Empty Simplices of Polytopes and Graded Betti Numbers

The conjecture of Kalai, Kleinschmidt, and Lee on the number of empty simplices of a simplicial polytope is established by relating it to the first graded Betti numbers of the polytope and applying a result of Migliore and the author. This approach allows us to derive explicit optimal bounds on the number of empty simplices of any given dimension. As a key result, we prove optimal bounds for the graded Betti numbers of any standard graded K-algebra in terms of its Hilbert function.     

Betti Numbers of Toric Varieties and Eulerian Polynomials

It is well known that the Eulerian polynomials, which count permutations in S _n by their number of descents, give the h-polynomial/h-vector of the simple polytopes known as permutohedra, the convex hull of the S _n -orbit for a generic weight in the weight lattice of S _n . Therefore, the Eulerian polynomials give the Betti numbers for certain smooth toric varieties associated with the permutohedra.

In this article we derive recurrences for the h-vectors of a family of polytopes generalizing this. The simple polytopes we consider arise as the orbit of a nongeneric weight, namely, a weight fixed by only the simple reflections J = {s _n , s _n?1, s _n?2,…, s _n?k+2, s _n?k+1} for some k with respect to the A _n root lattice. Furthermore, they give rise to certain rationally smooth toric varieties X(J) that come naturally from the theory of algebraic monoids. Using effectively the theory of reductive algebraic monoids and the combinatorics of simple polytopes, we obtain a recurrence formula for the Poincaré polynomial of X(J) in terms of the Eulerian polynomials.     

On the Betti Numbers of Chessboard Complexes

In this paper we study the Betti numbers of a type of simplicial complex known as a chessboard complex. We obtain a formula for their Betti numbers as a sum of terms involving partitions. This formula allows us to determine which is the first nonvanishing Betti number (aside from the 0-th Betti number). We can therefore settle certain cases of a conjecture of Björner, Lovász, Vreica, and ivaljevi in [2]. Our formula also shows that all eigenvalues of the Laplacians of the simplicial complexes are integers, and it gives a formula (involving partitions) for the multiplicities of the eigenvalues.     

Betti Numbers of Small Covers and Their Two-Fold Coverings

We compute the Betti numbers of two-fold coverings of small covers with some special properties; in particular, we use the results of Davis and Januszkiewicz on the cohomology of small covers. It turned out that their proof contains some gap that we describe in detail and fill in.