Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time

Peter Bürgisser (1) , Felipe Cucker (2) , Pierre Lairez (3)

1 Institut für Mathematik [Berlin]
2 HKU - Department of Mathematics
3 SPECFUN - Symbolic Special Functions : Fast and Certified

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of basic semialgebraic sets which works in weak exponential time. That is, out 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 which is doubly exponential (and this is so for almost all data).

Mots clés: Semialgebraic geometry; Theory of computation; Computing methodologies; Homology; Algorithm; Computational geometry; Hybrid symbolic-numeric methods

Domaines: Géométrie algorithmique [cs.CG]; Topologie algébrique [math.AT]; Géométrie algébrique [math.AG]

Peter Bürgisser, Felipe Cucker, Pierre Lairez. Computing the Homology of Basic Semialgebraic Sets in Weak Exponential Time. Journal of the ACM (JACM), 2018, 66 (1), pp.1-30. ⟨10.1145/3275242⟩. ⟨hal-01545657v2⟩