On Poisson customary polynomial identities

We study Poisson customary and Poisson extended customary polynomials. We show that the sequence of codimensions $\{r_n(V)\}_{n\geq1}$ of every extended customary space of variety $V$ of Poisson algebras over an arbitrary field is either bounded by a polynomial or at least exponential. Furthermore, if this sequence is bounded by polynomial then there is a polynomial $R(x)$ with rational coefficients such that $r_n(V)=R(n)$ for all sufficiently large $n$. We present lower and upper bounds for the polynomials $R(x)$ of an arbitrary fixed degree.