Persistent Homology of Semialgebraic Sets

Abstract

We give an algorithm with singly exponential complexity for computing the barcodes up to dimension (for any fixed ) of the filtration of a given semialgebraic set by the sublevel sets of a given polynomial. Our algorithm is the first algorithm for this problem with singly exponential complexity and generalizes the corresponding results for computing the Betti numbers up to dimension of semialgebraic sets with no filtration present.