On the computation of the topology of plane curves

Abstract

Let P ∈ Z[X, Y] be a square-free polynomial and C(P):= {(α, β) ∈ R², P(α, β) = 0} be the real algebraic curve defined by P. Our main result is an algorithm for the computation of the local topology in a neighbourhood of each of the singular points and critical points of the projection wrt the X-axis in Õ(d⁶τ+d⁷) bit operations where Õ means that we ignore logarithmic factors in d and τ. Compared to state of the art sub-algorithms used for computing a Cylindrical Algebraic Decomposition, this result avoids a generic shear and gives a deterministic algorithm for the computation of the topology of C(P) i.e a straight-line planar graph isotopic to C(P) in Õ(d⁶τ + d⁷) bit operations.