Betti Numbers Of Semi-algebraic Sets Research Articles

Polynomial Hierarchy, Betti Numbers, and a Real Analogue of Toda’s Theorem

Toda (in SIAM J. Comput. 20(5):865–877, 1991) proved in 1989 that the (discrete) polynomial time hierarchy, PH, is contained in the class P #P , namely the class of languages that can be decided by a Turing machine in polynomial time given access to an oracle with the power to compute a function in the counting complexity class #P. This result, which illustrates the power of counting, is considered to be a seminal result in computational complexity theory. An analogous result in the complexity theory over the reals (in the sense of Blum–Shub–Smale real machines in Bull. Am. Math. Soc. (NS) 21(1): 1–46, 1989) has been missing so far. In this paper we formulate and prove a real analogue of Toda’s theorem. Unlike Toda’s proof in the discrete case, which relied on sophisticated combinatorial arguments, our proof is topological in nature. As a consequence of our techniques, we are also able to relate the computational hardness of two extremely well-studied problems in algorithmic semi-algebraic geometry: the problem of deciding sentences in the first-order theory of the reals with a constant number of quantifier alternations, and that of computing Betti numbers of semi-algebraic sets. We obtain a polynomial time reduction of the compact version of the first problem to the second. This latter result may be of independent interest to researchers in algorithmic semi-algebraic geometry.

Approximation of definable sets by compact families, and upper bounds on homotopy and homology

We prove new upper bounds on homotopy and homology groups of o-minimal sets in terms of their approximations by compact o-minimal sets. In particular, we improve the known upper bounds on Betti numbers of semialgebraic sets defined by quantifier-free formulae, and obtain for the first time a singly exponential bound on Betti numbers of sub-Pfaffian sets.

Computing the Betti numbers of semi-algebraic sets defined by partly quadratic systems of polynomials

Let R be a real closed field, Q ⊂ R [ Y 1 , … , Y ℓ , X 1 , … , X k ] , with deg Y ( Q ) ⩽ 2 , deg X ( Q ) ⩽ d , Q ∈ Q , # ( Q ) = m , and P ⊂ R [ X 1 , … , X k ] with deg X ( P ) ⩽ d , P ∈ P , # ( P ) = s . Let S ⊂ R ℓ + k be a semi-algebraic set defined by a Boolean formula without negations, with atoms P = 0 , P ⩾ 0 , P ⩽ 0 , P ∈ P ∪ Q . We describe an algorithm for computing the Betti numbers of S generalizing a similar algorithm described in [S. Basu, Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (1) (2008) 45–80]. The complexity of the algorithm is bounded by ( ( ℓ + 1 ) ( s + 1 ) ( m + 1 ) ( d + 1 ) ) 2 O ( m + k ) . The complexity of the algorithm interpolates between the doubly exponential time bounds for the known algorithms in the general case, and the polynomial complexity in case of semi-algebraic sets defined by few quadratic inequalities [S. Basu, Computing the top few Betti numbers of semi-algebraic sets defined by quadratic inequalities in polynomial time, Found. Comput. Math. 8 (1) (2008) 45–80]. Moreover, for fixed m and k this algorithm has polynomial time complexity in the remaining parameters.

Errata for 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 $\sum_{i=0}^{\ell+2}{s\choose i}k^{2^{o(\min(\ell,s))}}$. For fixed l, the complexity of the algorithm can be expressed as $s^{\ell+2}+k^{2^{O(\ell)}}$, 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 $k^{2^{O(S)}}$.

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 $\sum_{i=0}^{\ell+2}{s\choose i}k^{2^{o(\min(\ell,s))}}$ . For fixed ℓ, the complexity of the algorithm can be expressed as $s^{\ell+2}+k^{2^{O(\ell)}}$ , 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 $k^{2^{O(S)}}$ .

Computing the first few Betti numbers of semi-algebraic sets in single exponential time

In this paper we describe an algorithm that takes as input a description of a semi-algebraic set S ⊂ R k , defined by a Boolean formula with atoms of the form P > 0 , P < 0 , P = 0 for P ∈ P ⊂ R [ X 1 , … , X k ] , and outputs the first ℓ + 1 Betti numbers of S , b 0 ( S ) , … , b ℓ ( S ) . The complexity of the algorithm is ( s d ) k O ( ℓ ) , where s = # ( P ) and d = max P ∈ P deg ( P ) , which is singly exponential in k for ℓ any fixed constant. Previously, singly exponential time algorithms were known only for computing the Euler–Poincaré characteristic, the zeroth and the first Betti numbers.

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.

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.

On the Betti numbers of semialgebraic sets defined by few quadratic inequalities

On the Betti numbers of semialgebraic sets defined by few quadratic inequalities