We survey both old and new developments in the theory of algorithms in real algebraic
geometry--starting from effective quantifier elimination in the first order theory of reals due to …

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 …

We introduce numerical algebraic geometry methods for computing lower bounds on the
reach, local feature size, and weak feature size of the real part of an equidimensional and …

Topological data analysis (TDA) provides tools for computing geometric and topological
information about spaces from a finite sample of points. We present an adaptive algorithm …

We describe and analyze an algorithm for computing the homology (Betti numbers and
torsion coefficients) of closed semialgebraic sets given by Boolean formulas without …

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 …

Let R be a real closed field. We prove that for each fixed ℓ, d≥ 0, there exists an algorithm
that takes as input a quantifier-free first-order formula Φ with atoms P= 0, P> 0, P< 0 with P∈ …

We extend the lower bounds on the complexity of computing Betti numbers proved in [P.
Bürgisser, F. Cucker, Counting complexity classes for numeric computations II: algebraic …

In this paper we describe a singly exponential algorithm for computing the first Betti number
of a given semi-algebraic set. Singly exponential algorithms for computing the zeroth Betti …

For any ℓ> 0, we present an algorithm which takes as input a semi-algebraic set, S, defined
by P 1≤ 0,…, P s≤ 0, where each P i∈ RX 1,…, X k has degree≤ 2, and computes the top …