# Log-Canonical Coordinates for Poisson Brackets and Rational Changes of Coordinates

## TL;DR

This paper proves that for affine space with a log-canonical Poisson bracket, no rational change of coordinates can make the bracket linear, establishing the minimal algebraic complexity of such coordinate systems.

## Contribution

It demonstrates that log-canonical coordinates are the simplest algebraic form for Poisson brackets on affine space, ruling out rational linearization.

## Key findings

- No rational change of coordinates can linearize log-canonical Poisson brackets.
- Identifies invariants of log-canonical Poisson brackets absent in linear brackets.
- Supports the idea that log-canonical coordinates are minimally complex.

## Abstract

Goodearl and Launois have shown that for a log-canonical Poisson bracket on affine space there is no rational change of coordinates for which the Poisson bracket is constant. Our main result is that if affine space is given a log-canonical Poisson bracket, then there does not exist any rational change of coordinates for which the Poisson bracket is linear. Hence, log-canonical coordinates can be thought of as the simplest possible algebraic coordinates for affine space with a log-canonical coordinate system. In proving this conjecture we find certain invariants of log-canonical Poisson brackets on affine space which linear Poisson brackets do not have.