Topology and arrangement computation of semi-algebraic planar curves

Topology and arrangement computation of semi-algebraic planar curves

We describe a new subdivision method to efficiently compute the topology and the arrangement of implicit planar curves. We emphasize that the output topology and arrangement are guaranteed to be correct. Although we focus on the implicit case, the algorithm can also treat parametric or piecewise linear curves without much additional work and no theoretical difficulties. The method isolates sing...

A Worst-case Bound for Topology Computation of Algebraic Curves

Computing the topology of an algebraic plane curve C means to compute a combinatorial graph that is isotopic to C and thus represents its topology in R2. We prove that, for a polynomial of degree n with coefficients bounded by 2ρ , the topology of the induced curve can be computed with Õ(n8(n+ρ2)) bit operations deterministically, and with Õ(n8ρ2) bit operations with a randomized algorithm in e...

Exact symbolic-numeric computation of planar algebraic curves

We present a novel certified and complete algorithm to compute arrangements of real planar algebraic curves. It provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition. From a high-level perspective, the overall method splits into two main subroutines, namely an algorithm denoted...

Elliptic Curves and Algebraic Topology

Elliptic curves enter algebraic topology through “Elliptic cohomology”–really a family of cohomology theories–and their associated “elliptic genera”. • Arithmetic aspect: Modularity of elliptic genera, The spectrum TMF of “topological modular forms” and the calculation of π∗TMF →MF (Z), Hopkins’s proof of Borcherds’ congruences. • Physical aspect: Witten’s approach to elliptic genera via string...

Topology of Plane Algebraic Curves