# Signatures of paths transformed by polynomial maps

## TL;DR

This paper provides a mathematical characterization of how the signature of a path changes under polynomial transformations, using algebraic structures like shuffle algebras and Zinbiel algebras.

## Contribution

It introduces a recursive algebra homomorphism that relates path signatures before and after polynomial transformations, generalizing previous results.

## Key findings

- Defines a path signature transformation via a shuffle algebra homomorphism
- Establishes the homomorphism's independence from specific paths
- Generalizes the main theorem using Zinbiel algebras

## 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. It allows us to describe the relation between the signature of a piecewise continuously differentiable path and the signature of the path obtained by transforming it under a polynomial map. We also study this map as a half-shuffle homomorphism and give a generalization of our main theorem in terms of Zinbiel algebras.