@article{cf45ee12efdd499d82d32792c471d492, title = "Computing the Homology of Semialgebraic Sets. I: Lax Formulas", 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. Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.", keywords = "Homology groups, Numerical algorithms, Weak complexity", author = "Peter B{\"u}rgisser and Felipe Cucker and Josu{\'e} Tonelli-Cueto", year = "2020", month = feb, doi = "10.1007/s10208-019-09418-y", language = "English", volume = "20", pages = "71--118", journal = "Foundations of Computational Mathematics", issn = "1615-3375", publisher = "Springer New York LLC", number = "1", }