Computing Path Signature Varieties in Macaulay2

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.