A Polynominal Time Algorithm for the Topological Type of a Real Algebraic Curve

Let I (:z: ,y ,z) be a homogeneous polynomial With. rational coefficients. Let C, be the real projective curve defined by ! = 0" and suppos~ that C/ is nonsingular. It is well known that C/ is essentially a finite collection of disjoint circles, all except possibly one of which lie in th~ projective plane Rp2 in such a way as to have both an interior (homeomorphic to a disk), and an exterior (homeomorphic to a Mobius strip).· These two-sided components of C/ are called ovals. The partial order im;posed on-its ovals by the relation of inclusion specifies the topological type ,or ·C/. We present an algorithm which, given I. determines the ordering of the ovals of C/. The algorithm constructs. a cell complex f.or IRp2, such that for each oval 0 of C/, the closure of each component of complement(0) is a. subcomplex. The Euler characteristic X of a complex is easily computed, and since X(disk)¢X(MBbius strip), any cell can be classified as being inside, on, or outside a particular oval. This essentially determines the ordering of ovals. The maximum computing time of our algoriUun is dominated by '8 polynomial rW?-ction or the degree of I and th~ size of its coefficients.