The purpose of this paper is to give a characterisation of divided power algebras over a reduced operad. Such a characterisation is given in terms of polynomial operations, following the classical example of divided power algebras.We describe these polynomial operations in two different ways: one way uses invariant elements under the action of the symmetric group, the other coinvariant elements. Our results are then applied to the case of level algebras, which are (non-associative) commutative algebras satisfying the exchange law.• In the set Π(r; n) there is a unique element of Comp p (n) (more precisely, in the image of Comp p (n) by ι), namely r, and Π p (n) = r∈Comp p (n) Π(r; n).