The Rough Veronese variety

TL;DR

This paper investigates the algebraic structure of signature tensors of rough paths, revealing that the associated signature variety, called the Rough Veronese, is toric and has complex ideal generation properties.

Contribution

It proves that the Rough Veronese variety is toric and addresses the ideal generation question, extending understanding of signature varieties for rough paths.

Findings

The Rough Veronese variety is toric.

The universal signature variety's ideal is not generated solely by quadrics.

Provides new algebraic tools for studying rough path signatures.

Abstract

We study signature tensors of paths from a geometric viewpoint. The signatures of a given class of paths parametrize an algebraic 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.