Topology of real algebraic space curves

Topology of real algebraic space curves

In this paper we give a new projection-based algorithm for computing the topology of a real algebraic space curve given implicitly by a set of equations. Under some genericity conditions, which may be reached through a linear change of coordinates, we show that a plane projection of the given curve, together with a special polynomial in the ideal of the curve contains all the information needed...

On the Topology of Real Algebraic Plane Curves

We revisit the problem of computing the topology and geometry of a real algebraic plane curve. The topology is of prime interest but geometric information, such as the position of singular and critical points, is also relevant. A challenge is to compute efficiently this information for the given coordinate system even if the curve is not in generic position. Previous methods based on the cylind...

Topology of Plane Algebraic Curves

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...

Real Plane Algebraic Curves

We study real algebraic plane curves, at an elementary level, using as little algebra as possible. Both cases, affine and projective, are addressed. A real curve is infinite, finite or empty according to the fact that a minimal polynomial for the curve is indefinite, semi–definite nondefinite or definite. We present a discussion about isolated points. By means of the operator, these points can ...