Cellular decompositions and Chebyshev interpolants for real algebraic curves

A cellular decomposition of a real algebraic curve consists of a collection of vertices and edges which have a smooth interior. A numerical cellular decomposition represents an edge via one interior point and a homotopy that permits tracking along the edge. This homotopy yields a parameterization of the edge that can be used for performing various operations such as membership testing, computing winding numbers at each boundary vertex, and generating sample points. By combining numerical cellular decomposition with Chebyshev interpolants along each edge, we develop an approach that can efficiently perform computations such as optimizing an analytic or nonanalytic function over real algebraic curves. Examples utilizing Bertini real to compute a numerical cellular decomposition and Chebfun to compute Chebyshev interpolants are provided.