  In this note we introduce higher order polar loci as natural generalizations
of the classical polar loci, replacing the role of tangent spaces by that of
higher order osculating spaces. The close connection between polar loci and
dual varieties carries over to a connection between higher order polar loci and
higher order dual varieties. We generalize the duality between the degrees of
polar classes of a variety and those of its dual variety to varieties that are
reflexive to a higher order. In particular, the degree of the top (highest
codimension) polar class of order k is equal to the degree of the k-th dual
variety. We define higher order Euclidean normal bundles and use them to define
higher order reciprocal polar loci and classes. We give examples of how to
compute the degrees of the higher order polar and reciprocal polar classes in
some special cases: curves, scrolls, and toric varieties.

Algebraic Geometry