Equivalence classes of planar algebraic curves through numerical algebraic geometry

For the action of a group on the plane, the group equivalence problem for curves can be stated as: given two curves, decide if they are related by an element of the group. We describe an efficient equality test, using tools from numerical algebraic geometry, to determine (with "probability-one") whether or not two rational maps have the same image up to Zariski closure. Using signature maps, constructed from differential and joint invariants, we apply this test to solve the group equivalence problem for algebraic curves under the linear action of algebraic groups. In this talk I will discuss the equality test and signature maps for algebraic curves, focusing on the action of the complex Euclidean group for our computations and examples. I will present some of our results comparing the sensitivity of different signature maps. This is based on joint work with Tim Duff.