Computing Path Signature Varieties in Macaulay2

TL;DR

This paper introduces a Macaulay2 package called PathSignatures that facilitates the computation and algebraic study of path signature tensors for piecewise polynomial paths, aiding research in algebraic geometry and path analysis.

Contribution

The paper presents a new Macaulay2 package that simplifies the computation and analysis of path signature varieties for piecewise polynomial paths.

Findings

Efficient computation of signature tensors using the package.

Algebraic description of signature varieties.

Tools for manipulating parametrized path families.

Abstract

The signature of a path is a non-commutative power series whose coefficients are given by certain iterated integrals over the path coordinates. This series almost uniquely characterizes the path up to translation and reparameterization. Taking only fixed degree parts of these series yields signature tensors. We introduce the Macaulay2 package $\texttt{PathSignatures}$ to simplify the study of these interesting objects for piecewise polynomial paths. It allows for the creation and manipulation of parametrized families of paths and provides methods for computing their signature tensors and their associated algebraic varieties.