Computing the Homology of Semialgebraic Sets. II: General Formulas

Bürgisser, Peter; Cucker, Felipe; Tonelli-Cueto, Josué

doi:10.1007/s10208-020-09483-8

Cited by 11 publications

(15 citation statements)

References 20 publications

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“…We note here that designing algorithms with singly exponential complexity has being a leit motif in the research in algorithmic semi-algebraic geometry over the past decades -starting from the so called "critical-point method" which resulted in algorithms for testing emptiness, connectivity, computing the Euler-Poincaré characteristic, as well as for the first few Betti numbers of semi-algebraic sets (see [2] for a history of these developments and contributions of many authors). More recently, such algorithms have also been developed in other (more numerical) models of computations [11,12,13].…”

Section: Introductionmentioning

confidence: 99%

“…In [11,12,13], the authors take a different approach. Working over R, and given a "well-conditioned" semi-algebraic subset S Ă R k , they compute a witness complex whose geometric realization is k-equivalent to S. The size of this witness complex as well the complexity of the algorithm is bounded singly exponentially in k, but also depends on a real parameter, namely the condition number of the input (and so this bound is not uniform).…”

Section: Introductionmentioning

confidence: 99%

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Computing the homology functor on semi-algebraic maps and diagrams

Basu¹,

Karisani²

2022

Preprint

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Developing an algorithm for computing the Betti numbers of semi-algebraic sets with singly exponential complexity has been a holy grail in algorithmic semi-algebraic geometry and only partial results are known. In this paper we consider the more general problem of computing the image under the homology functor of a semi-algebraic map f : X Ñ Y between closed and bounded semi-algebraic sets. For every fixed ě 0 we give an algorithm with singly exponential complexity that computes bases of the homology groups H i pXq, H i pY q (with rational coefficients) and a matrix with respect to these bases of the induced linear maps H i pf q : H i pXq Ñ H i pY q, 0 ď i ď . We generalize this algorithm to more general (zigzag) diagrams of maps between closed and bounded semi-algebraic sets and give a singly exponential algorithm for computing the homology functors on such diagrams. This allows us to give an algorithm with singly exponential complexity for computing barcodes of semi-algebraic zigzag persistent homology in small dimensions.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Computing the homology functor on semi-algebraic maps and diagrams

Basu¹,

Karisani²

2022

Preprint

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“…The reciprocal of the reach, called the condition number of the underlying system of polynomials, is also a very important indicator, see for instance [4,5]. Here the authors derive a complexity analysis of an algorithm for computing homology, using condition numbers.…”

Section: Introductionmentioning

confidence: 99%

The critical curvature degree of an algebraic variety

Horobet¹

2021

Preprint

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“…if the condition number is infinite). In this latter model, algorithms with singly exponential complexity for computing all the Betti numbers of semi-algebraic sets have been developed [14,15,16]. As noted above, these algorithms will fail to produce any result on certain inputs.…”

Section: References 29 1 Introductionmentioning

confidence: 99%

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

Basu,

Percival

2021

Preprint

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Let R be a real closed field and C the algebraic closure of R. We give an algorithm for computing a semi-algebraic basis for the first homology group, H1(S, F), with coefficients in a field F, of any given semi-algebraic set S ⊂ R k defined by a closed formula. The complexity of the algorithm is bounded singly exponentially. More precisely, if the given quantifier-free formula involves s polynomials whose degrees are bounded by d, the complexity of the algorithm is bounded by (sd. This algorithm generalizes well known algorithms having singly exponential complexity for computing a semi-algebraic basis of the zero-th homology group of semi-algebraic sets, which is equivalent to the problem of computing a set of points meeting every semi-algebraically connected component of the given semi-algebraic set at a unique point. It is not known how to compute such a basis for the higher homology groups with singly exponential complexity.As an intermediate step in our algorithm we construct a semi-algebraic subset Γ of the given semi-algebraic set S, such that Hq(S, Γ) = 0 for q = 0, 1. We relate this construction to a basic theorem in complex algebraic geometry stating that for any affine variety X of dimension n, there exists Zariski closed subsetswith dimC Z (i) ≤ i, and Hq(X, Z (i) ) = 0 for 0 ≤ q ≤ i. We conjecture a quantitative version of this result in the semi-algebraic category, with X and Z (i) replaced by closed semi-algebraic sets. We make initial progress on this conjecture by proving the existence of Z (0) and Z (1) with complexity bounded singly exponentially (previously, such an algorithm was known only for constructing Z0).

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Computing the Homology of Semialgebraic Sets. II: General Formulas

Cited by 11 publications

References 20 publications

Computing the homology functor on semi-algebraic maps and diagrams

Computing the homology functor on semi-algebraic maps and diagrams

The critical curvature degree of an algebraic variety

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

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