Computing the Homology of Semialgebraic Sets. I: Lax Formulas

  • Peter Bürgisser
  • Felipe CuckerEmail author
  • Josué Tonelli-Cueto


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.


Homology groups Weak complexity Numerical algorithms 

Mathematics Subject Classification

14P10 65D18 65Y20 68Q25 



We are grateful to Saugata Basu, Pierre Lairez, and Nicolai Vorobjov for helpful discussions.


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Copyright information

© SFoCM 2019

Authors and Affiliations

  • Peter Bürgisser
    • 1
  • Felipe Cucker
    • 2
    Email author
  • Josué Tonelli-Cueto
    • 1
  1. 1.Institut für MathematikTechnische Universität BerlinBerlinGermany
  2. 2.Department of MathematicsCity University of Hong KongKowloon TongHong Kong

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