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Free Poisson algebras are very closely connected with polynomial algebras, and the Poisson brackets are used to solve many problems in affine algebraic geometry. In this note, we study Poisson derivations on the symplectic Poisson algebra, and give a connection between the Jacobian conjecture with derivations on the symplectic Poisson algebra.

Let $k\subseteq k^{\text{'}}$ be a field extension. We give relations between the kernels of higher derivations on $k\left[X\right]$ and $k^{\text{'}}\left[X\right]$, where $k\left[X\right]:=k[x_1,\cdots ,x_n]$ denotes the polynomial ring in $n$ variables over the field $k$. More precisely, let $D={\left\{D_n\right\}}_{n=0}^{\infty }$ a higher $k$-derivation on $k\left[X\right]$ and $D^{\text{'}}={\left\{D_n^{\text{'}}\right\}}_{n=0}^{\infty }$ a higher $k^{\text{'}}$-derivation on $k^{\text{'}}\left[X\right]$ such that $D_m^{\text{'}}\left(x_i\right)=D_m\left(x_i\right)$ for all $m\ge 0$ and $i=1,2,\cdots ,n$. Then (1) $k{\left[X\right]}^D=k$ if and only if $k^{\text{'}}{\left[X\right]}^{D^{\text{'}}}=k^{\text{'}}$; (2) $k{\left[X\right]}^D$ is a finitely generated $k$-algebra if and only if $k^{\text{'}}{\left[X\right]}^{D^{\text{'}}}$ is a finitely generated $k^{\text{'}}$-algebra. Furthermore, we also show that the kernel $k{\left[X\right]}^D$ of a higher derivation $D$ of $k\left[X\right]$ can be generated by a set...