  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.
