# Hoon Hong

Department of Mathematics North Carolina State University USA

**Connectivity in Semialgebraic Sets**

**= Symbolic and Numeric Approach=**

*Hoon Hong*

Department of Mathematics

North Carolina State University

USA

**Abstract.** A semialgebraic set is a subset of real space defined by polynomial equations and inequalities. A semialgebraic set is a union of finitely many maximally connected components. In this talk, we consider the problem of deciding whether two given points in a semialgebraic set are connected, that is, whether the two points lie in a same connected component. In particular, we consider the semialgebraic set defined by f not equal 0 where f is a given bivariate polynomial. The motivation comes from the observation that many important/non-trivial problems in science and engineering can be often reduced to that of connectivity. Due to it importance, there has been intense research effort on the problem. We will describe a method based on gradient fields and provide a sketch of the proof of correctness based Morse complex. The method seems to be more efficient than the previous methods in practice.