This is a description of joint work with Irina Kogan and Cynthia Vinzant.

- Complete results are available on the arXiv here.

- Slides are available here for a 2019 seminar talk on this topic at MPI-MiS Leipzig.

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Deciding when two objects are equivalent under a particular group of transformations is an important problem with applications to areas such as image science and differential equations. One can determine equivalence of smooth structures by examining their images under a suitable choice of differential invariants for the group action. In the case of smooth curves under the smooth action of a group on the plane, one can associate a signature curve, which characterizes equivalence classes of curves under this action. Signature curves have been used in various applications including medical imaging and jigsaw puzzle reconstruction.

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When the curves are algebraic, the implicit equation of the signature can be computed using Gröbner basis algorithms, and the signature can be studied and applied using computational algebraic geometry. In this way, one can connect differential invariants and equivalence classes and invariants of algebraic curves, a topic of classical algebraic geometry and invariant theory.

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While elimination algorithms give a straightforward method to compute signature polynomials, they are computationally expensive in practice. For curves under subgroups of the projective action, we examine properties of signature polynomials that one could determine without their explicit computation, in particular their degree.

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Below is a link for Maple code (and a viewable pdf of the worksheet) to generate signature polynomials for algebraic curves under select algebraic groups and to explore their properties. The supplementary materials for the paper are also available below.

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Exploring Signature Polynomials Viewable Worksheet

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Supplementary Material for Differential Signatures of Algebraic Curves