The rough Veronese variety
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Submission date: 19. Oct. 2018
published in: Linear algebra and its applications, 583 (2019), p. 282-299
DOI number (of the published article): 10.1016/j.laa.2019.08.029
MSC-Numbers: 14Q15, 14M25, 60H99
Keywords and phrases: signature tensor, rough path, Lyndon words
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We study signature tensors of paths from an algebraic geometric viewpoint. The signatures of a given class of paths parametrize a variety inside the space of tensors, and these signature varieties provide both new tools to investigate paths and new challenging questions about their behavior. This paper focuses on signatures of rough paths. Their signature variety shows surprising analogies with the Veronese variety, and our aim is to prove that this so-called Rough Veronese is toric. The same holds for the universal variety. Answering a question of Amendola, Friz and Sturmfels, we show that the ideal of the universal variety does not need to be generated by quadrics.