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Computing the Homology of Semialgebraic Sets I: Lax Formulas
Zitatschlüssel PFJ-Computing-The-Homology-Of-Semialgebraic-Sets-I-Lax-Formulas
Autor Bürgisser, Peter and Cucker, Felipe and Tonelli-Cueto, Josué
Seiten 71–118
Jahr 2020
DOI 10.1007/s10208-019-09418-y
Journal Foundations of Computational Mathematics
Jahrgang 20
Nummer 1
Zusammenfassung 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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