  The quadratic Poisson Gel'fand-Kirillov problem asks whether the field of
fractions of a Poisson algebra is Poisson birationally equivalent to a Poisson
affine space, i.e. to a polynomial algebra $\K[X_1,..., X_n]$ with Poisson
bracket defined by $\{X_i,X_j\}=\lambda_{ij} X_iX_j$ for some skew-symmetric
matrix $(\lambda_{ij}) \in M_n(\K)$. This problem was studied in \cite{GL} over
a field of characteristic 0 by using a Poisson version of the
deleting-derivations algorithm of Cauchon. In this paper, we study the
quadratic Poisson Gel'fand-Kirillov problem over a field of arbitrary
characteristic. In particular, we prove that the quadratic Poisson
Gel'fand-Kirillov problem is satisfied for a large class of Poisson algebras
arising as semiclassical limits of quantised coordinate rings. For, we
introduce the concept of {\it higher Poisson derivation} which allows us to
extend the Poisson version of the deleting-derivations algorithm from the
characteristic 0 case to the case of arbitrary characteristic.
  When a torus is acting rationally by Poisson automorphisms on a Poisson
polynomial algebra arising as the semiclassical limit of a quantised coordinate
ring, we prove (under some technical assumptions) that quotients by Poisson
prime torus-invariant ideals also satisfy the quadratic Poisson
Gel'fand-Kirillov problem. In particular, we show that coordinate rings of
determinantal varieties satisfy the quadratic Poisson Gel'fand-Kirillov
problem.
