Log-canonical coordinates for Poisson brackets and rational changes of coordinates

Goodearl and Launois have shown in Goodearl and Launois (2011) 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 a proof of a conjecture of Michael Shapiro which states 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.