Computing the First Betti Numberand Describing the Connected Components of Semi-algebraic Sets

In this paper we describe a singly exponential algorithm for computing the first Betti number of a given semi-algebraic set. Singly exponential algorithms for computing the zero-th Betti number, and the Euler-Poincar\'e characteristic, were known before. No singly exponential algorithm was known for computing any of the individual Betti numbers other than the zero-th one. We also give algorithms for obtaining semi-algebraic descriptions of the semi-algebraically connected components of any given real algebraic or semi-algebraic set in single-exponential time improving on previous results

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 $S \subset \R^k$, defined by a Boolean formula with atoms of the form $P > 0, P < 0, P=0$ for $P \in {\mathcal P} \subset \R[X_1,...,X_k],$ and outputs the first $\ell+1$ Betti numbers of $S$, $b_0(S),...,b_\ell(S).$ The complexity of the algorithm is $(sd)^{k^{O(\ell)}},$ where where s = #({\mathcal P}) and $d = \max_{P\in {\mathcal P}}{\rm deg}(P),$ which is singly exponential in $k$ for $\ell$ any fixed constant. Previously, singly exponential time algorithms were known only for computing the Euler-Poincar\'e characteristic, the zero-th and the first Betti numbers

Efficient algorithms for computing the Euler-Poincar\'e characteristic of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincar\'e characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincar\'e characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.Comment: 29 pages, 1 Figure. arXiv admin note: substantial text overlap with arXiv:1312.658

Efficient simplicial replacement of semi-algebraic sets and applications

We prove that for any $\ell \geq 0$, there exists an algorithm which takes as input a description of a semi-algebraic subset $S \subset \mathbb{R}^k$ given by a quantifier-free first order formula $\phi$ in the language of the reals, and produces as output a simplicial complex $\Delta$, whose geometric realization, $|\Delta|$ is $\ell$-equivalent to $S$. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell)}}$, where $s$ is the number of polynomials appearing in the formula $\phi$, and $d$ a bound on their degrees. For fixed $\ell$, this bound is \emph{singly exponential} in $k$. In particular, since $\ell$-equivalence implies that the \emph{homotopy groups} up to dimension $\ell$ of $|\Delta|$ are isomorphic to those of $S$, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell$ homotopy groups of $S$ to the combinatorial problem of computing the first $\ell$ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell)}}$. As an application we give an algorithm with singly exponential complexity for computing the \emph{persistence barcodes} up to dimension $\ell$ (for any fixed $\ell \geq 0$), of the filtration of a given semi-algebraic set by the sub-level sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity, and generalizes the corresponding results for computing the Betti numbers up to dimension $\ell$ of semi-algebraic sets with no filtration present.Comment: 62 pages, 6 figure

Bounding the number of stable homotopy types of a parametrized family of semi-algebraic sets defined by quadratic inequalities

We prove a nearly optimal bound on the number of stable homotopy types occurring in a k-parameter semi-algebraic family of sets in $\R^\ell$, each defined in terms of m quadratic inequalities. Our bound is exponential in k and m, but polynomial in $\ell$. More precisely, we prove the following. Let $\R$ be a real closed field and let $${\mathcal P} = \{P_1,...,P_m\} \subset \R[Y_1,...,Y_\ell,X_1,...,X_k],$$ with ${\rm deg}_Y(P_i) \leq 2, {\rm deg}_X(P_i) \leq d, 1 \leq i \leq m$. Let $S \subset \R^{\ell+k}$ be a semi-algebraic set, defined by a Boolean formula without negations, whose atoms are of the form, $P \geq 0, P\leq 0, P \in {\mathcal P}$. Let $\pi: \R^{\ell+k} \to \R^k$ be the projection on the last k co-ordinates. Then, the number of stable homotopy types amongst the fibers S_{\x} = \pi^{-1}(\x) \cap S is bounded by $$(2^m\ell k d)^{O(mk)}.$$Comment: 27 pages, 1 figur