Computing the Homology of Semialgebraic Sets. II: General Formulas

We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of th...

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Bibliographic Details
Published inFoundations of computational mathematics Vol. 21; no. 5; pp. 1279 - 1316
Main Authors Bürgisser, Peter, Cucker, Felipe, Tonelli-Cueto, Josué
Format Journal Article
LanguageEnglish
Published New York Springer US 01.10.2021
Springer
Springer Nature B.V
Springer Verlag
Subjects
Algebraic Geometry
Algorithms
Applications of Mathematics
Boolean algebra
Computation
Computational Geometry
Computer Science
Economics
Homology
Linear and Multilinear Algebras
Math Applications in Computer Science
Mathematics
Mathematics and Statistics
Matrix Theory
Numerical Analysis
68Q25
Semialgebraic sets
Homology groups
Numerical algorithms
14P10
65Y20
Weak complexity
65D18
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Summary:We describe and analyze a numerical algorithm for computing the homology (Betti numbers and torsion coefficients) of semialgebraic sets given by Boolean formulas. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. This extends the work in Part I to arbitrary semialgebraic sets. All previous algorithms proposed for this problem have doubly exponential complexity.
ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-020-09483-8