Abstract
We characterize the signature of piecewise continuously differentiable paths transformed by a polynomial map in terms of the signature of the original path. For this aim, we define recursively an algebra homomorphism between two shuffle algebras on words. This homomorphism does not depend on the path and behaves well with respect to composition and homogeneous maps. We also study this map as a half-shuffle homomorphism and give a generalization of our main theorem in terms of Zinbiel algebras.
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Acknowledgements
The authors would like to thank Bernd Sturmfels, Francesco Galuppi, Joscha Diehl, Mateusz Michałek, and Max Pfeffer for their fruitful conversations and observations. The collaboration between the co-authors would not have been possible without the financial support from the research institute MPI-MiS Leipzig (Germany).
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R. Preiß is currently supported by European Research Council through CoG-683164.
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Colmenarejo, L., Preiß, R. Signatures of paths transformed by polynomial maps. Beitr Algebra Geom 61, 695–717 (2020). https://doi.org/10.1007/s13366-020-00493-9
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DOI: https://doi.org/10.1007/s13366-020-00493-9