On interactions of massless integer high spin and scalar fields

We apply an unconstrained formulation of bosonic higher spin fields to study interactions of these fields with a bosonic field using new method for the deformation procedure. It is proved that local vertices of any order containing one higher spin $s$ field and arbitrary number of scalar fields and being invariant under original gauge transformations are described with the help of one local totally symmetric $(s-2)$-rank tensor of scalar fields. This tensor is explicitly constructed for particular cases related to cubic vertices for spin $s$ and vertices of an arbitrary order for spin $s=4$.