The critical curvature degree of an algebraic variety

Abstract

In this article we study the complexity involved in the computation of the reach in arbitrary dimension and in particular the computation of the critical spherical curvature points of an arbitrary algebraic variety. We present properties of the critical spherical curvature points as well as an algorithm for computing it.

Enumerative Geometry of Curvature of Algebraic Hypersurfaces

“…In this case, the coefficients of x 2 1 x 2 2 and x 2 2 x 2 3 are positive, while the coefficient of x 2 1 x 2 3 is negative. We can therefore write (18) as a sum-of-squares:…”

“…Recent work on counting curvature points includes [3,18,26]. The papers [3] and [26] address critical curvature in the case of plane curves, describing a correspondence between the critical curvature points on a plane curve and the cusps on its evolute.…”

“…The possible number of real solutions is not known (see the discussion around [26,Problem 2]). The paper [18] generalizes these ideas to varieties of higher dimension, but takes a different approach than ours.…”