Persistent homology of semi-algebraic sets

Abstract

We give an algorithm with singly exponential complexity for computing the barcodes up to dimension (for any fixed ≥ 0), of the filtration of a given semi-algebraic set by the sub-level 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 semi-algebraic sets with no filtration present.