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Computing the Homology of Semialgebraic Sets I: Lax Formulas
Citation key PFJ-Computing-The-Homology-Of-Semialgebraic-Sets-I-Lax-Formulas
Author Bürgisser, Peter and Cucker, Felipe and Tonelli-Cueto, Josué
Pages 71–118
Year 2020
DOI 10.1007/s10208-019-09418-y
Journal Foundations of Computational Mathematics
Volume 20
Number 1
Abstract We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. 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. All previous algorithms solving this problem have doubly exponential complexity (and this is so for almost all input data). Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.
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