Efficient simplicial replacement of semi-algebraic sets

Abstract

Designing an algorithm with a singly exponential complexity for computing semi-algebraic triangulations of a given semi-algebraic set has been a holy grail in algorithmic semi-algebraic geometry. More precisely, given a description of a semi-algebraic set S ⊂ R k by a first order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex ∆, whose geometric realization, |∆|, is semi-algebraically homeomorphic to S. In this paper we consider a weaker version of this question. We prove that for any ≥ 0, there exists an algorithm which takes as input a description of a semi-algebraic subset S ⊂ R k given by a quantifier-free first order formula φ in the language of the reals, and produces as output a simplicial complex ∆, whose geometric realization, |∆| is -equivalent to S. The complexity of our algorithm is bounded by (sd) k O( ) , where s is the number of polynomials appearing in the formula φ, and d a bound on their degrees. For fixed , this bound is singly exponential in k. In particular, since -equivalence implies that the homotopy groups up to dimension of |∆| are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first homotopy groups of S to the combinatorial problem of computing the first homotopy groups of a finite simplicial complex of size bounded by (sd) k O( ) .As an application we give an algorithm with singly exponential complexity for computing the persistence barcodes up to dimension (for any fixed ≥ 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 of semi-algebraic sets with no filtration present.

Persistent homology of semi-algebraic sets

“…We refer the reader to [8] for the details. The complexity of this algorithm, as well as the size of the output simplicial complex ∆, are bounded by…”

“…We remark that it is plausible that after ensuring the finiteness of the filtration, the last step of computing the barcode could be achieved by an appropriate extension of the algorithm for computing the first few Betti numbers of semialgebraic sets described in [6]. However, this extension would be non-trivial and we prefer to use directly Algorithm 3 in [8] for which no extension is needed.…”

Efficient computation of a semi-algebraic basis of the first homology group of a semi-algebraic set

“…For example, there are algorithms with singly exponential complexity for computing the first Betti numbers (for any fixed ≥ 0) which do not use triangulations [2]. More recently, in [9], the authors give an algorithm with singly exponential complexity for computing a simplicial complex which is homologically -equivalent (see Definition 9 below) to a given closed semi-algebraic set (for any fixed ≥ 0).…”

“…The standard algorithms for triangulating semi-algebraic sets using cylindrical algebraic decomposition (see for example [6, Chapter 5]) can be modified so that their complexities are bounded doubly exponentially only in the dimension of the given semi-algebraic set rather than that of the ambient space. Hence if Conjecture 1 is true, then it will provide an alternative approach (compared to [2,9]) towards the problem of computing the the first Betti numbers of any given semi-algebraic set with singly exponential complexity for each fixed . In this paper, we take an initial step towards verifying the conjecture (see Theorem 3).…”

“…For this purpose we use an algorithm for replacing a given semialgebraic set and a tuple of subsets by a simplicial complex and a tuple of corresponding subcomplexes, which are homologically -equivalent (cf. Definition 9) for any fixed , and which has singly exponentially bounded complexity [9] (cf. Algorithm 6 below).…”