Polar varieties are central objects in algebraic geometry. Every subvariety in a projective space has an associated list of polar varieties, encoding its tangential properties. Their degrees determine codimension and degree of the dual variety. Moreover, under mild generality assumptions, the polar degrees sum up to the Euclidean distance degree. This quantity is the algebraic degree of the distance of the given variety to a generic point in projective space. It plays an important role in the context of variety learning and algebraic sampling. Recently it has been shown that the polar degrees of a projective variety coincide with the degrees of its coisotropic hypersurfaces. These hypersurfaces live inside Grassmannians and appear naturally in computer vision.
Many of the above objects have been generalized to higher order analogues. Our goal is to extend this generalization to polar geometry to capture higher tangency properties of projective varieties. Projective duality has been expanded to higher order duality by allowing higher order contact, called osculation. Coisotropic hypersurfaces have been generalized to coisotropic varieties, which have arbitrary codimension in their ambient Grassmannian. We will introduce a new notion of higher order polar varieties to create the missing link between higher order duality and coisotropic varieties. We will also study higher order Euclidean distance degrees, describe our new concepts especially for toric varieties, and analyze their tropicalizations.
This project is foundational research within algebraic geometry with a view towards computations and applications in computer vision and algebraic sampling. In addition to algebro-geometric methods (such as intersection theory or the study of resultants and discriminants) it requires techniques from a variety of other disciplines, such as combinatorics, convex geometry, statistics, computer vision, tropical geometry, and both symbolic and numerical computations.