Since a real univariate polynomial does not always have real roots, a very natural
algorithmic problem, is to design a method to count the number of real roots of a given …

George Collins' discovery of Cylindrical Algebraic Decomposition (CAD) as a method for
Quantifier Elimination (QE) for the elementary theory of real closed fields brought a major …

The arrangement of a finite collection of geometric objects is the decomposition of the space
into connected cells induced by them. We survey combinatorial and algorithmic properties of …

We consider a semi-algebraic set $ S $ defined by $ s $ polynomials in $ k $ variables which
is contained in an algebraic variety $ Z (Q) $. The variety is assumed to have real dimension …

We study several polygonal curve problems under the Fréchet distance via algebraic
geometric methods. Let 𝕏 dm and 𝕏 dk be the spaces of all polygonal curves of m and k …

In this paper we present approximate algorithms for matching two polygonal curves with
respect to the Fréchet distance. We define a discrete version of the Fréchet distance as a …

study the motion-planning problem for pairs and tripIes of robots operating in a shared
workspace containing n obstacles. A standard way to solve such problems is to view the …

Let R be a real closed field, P,Q⊂RX_1,...,X_k finite subsets of polynomials, with the
degrees of the polynomials in P (resp., Q) bounded by d (resp., d 0). Let V⊂R^k be the real …

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 …

Topological complexity of semialgebraic sets in Rk has been studied by many researchers
over the past fifty years. An important measure of the topological complexity are the Betti …