Differential Signatures of Algebraic Curves

In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a plane algebraic curve another plane algebraic curve (a signature curve) in such a way that two generic curves have the same signatures if and only if they are $G$-equivalent. We prove that for any $G$-action, there exists a pair of rational differential invariants, called classifying invariants, that can be used to construct signatures. We derive a formula for the degree of a signature curve in terms of the degree of the original curve, the size of its symmetry group, and some quantities depending on a choice of classifying invariants. We show that all generic curves have signatures of the same degree, and this degree is the sharp upper bound. For the full projective group, as well as for its affine, special affine, and special Euclidean subgroups, we give explicit sets of rational classifying invariants and derive a formula for the degree of the signature curve of a generic curve as a quadratic function of the degree of the original curve.