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The Veronese subalgebra $A_0$ of degree $d\geq 2$ of the polynomial algebra$A=K[x_1,x_2,\ldots,x_n]$ over a field $K$ in the variables$x_1,x_2,\ldots,x_n$ is the subalgebra of $A$ generated by all monomials ofdegree $d$ and the Veronese subalgebra $P_0$ of degree $d\geq 2$ of the freePoisson algebra $P=P\langle x_1,x_2,\ldots,x_n\rangle$ is the subalgebraspanned by all homogeneous elements of degree $kd$, where $k\geq 0$. If $n\geq2$ then every derivation and every locally nilpotent derivation of $A_0$ and$P_0$ over a field $K$ of characteristic zero is induced by a derivation and alocally nilpotent derivation of $A$ and $P$, respectively. Moreover, we provethat every automorphism of $A_0$ and $P_0$ over a field $K$ closed with respectto taking all $d$-roots of elements is induced by an automorphism of $A$ and$P$, respectively.