# Differential signatures of algebraic curves

## TL;DR

This paper develops a method to classify complex algebraic curves under projective transformations using differential invariants, providing explicit formulas for signature curves and their degrees, enhancing curve equivalence analysis.

## Contribution

It introduces rational differential invariants for group actions on algebraic curves and derives explicit degree formulas for signature curves under various subgroups.

## Key findings

- Existence of rational classifying invariants for G-actions.
- Degree of signature curves expressed as quadratic functions of original curve degree.
- Explicit formulas for signature degrees under projective and subgroups.

## Abstract

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. 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. We show that this generic degree is the sharp upper bound.