Automorphisms of Veronese subalgebras of polynomial algebras and free Poisson algebras

The Veronese subalgebra A(0) of degree d >= of the polynomial algebra A = [x(1),x(2),...,x(n)] over a field K in the variables x(1), x(2), . . . ,x(n) is the subalgebra of A generated by all monomials of degree d and the Veronese subalgebra P-0 of degree d >= 2 of the free Poisson algebra P = P < x(1), x(2), . . . , x(n)> is the subalgebra spanned by all homogeneous elements of degree kd, where k >= 0. If n >= 2 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 a locally nilpotent derivation of A and P, respectively. Moreover, we prove that every automorphism of A(0) and P-0 over a field K closed with respect to taking all d-roots of elements is induced by an automorphism of A and P, respectively.