# Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

```@article{Cucker2019ComputingTH,
title={Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time},
author={Felipe Cucker and Peter B{\"u}rgisser and Pierre Lairez},
journal={Journal of the ACM (JACM)},
year={2019},
volume={66},
pages={1 - 30}
}```
• Published 22 June 2017
• Computer Science, Mathematics
• Journal of the ACM (JACM)
We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets that works in weak exponential time. That is, of a set of exponentially small measure in the space of data, the cost of the algorithm is exponential in the size of the data. All algorithms previously proposed for this problem have a complexity that is doubly exponential (and this is so for almost all data).
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