  Assuming locality, Lorentz invariance and parity conservation we obtain a set
of differential equations governing the 3-point interactions of massless
bosons, which in turn determines the polynomial ring of these amplitudes. We
derive all possible 3-point interactions for tensor fields with polarisations
that have total symmetry and mixed symmetry under permutations of Lorentz
indices. Constraints on the existence of gauge-invariant cubic vertices for
totally symmetric fields are obtained in general spacetime dimensions and are
compared with existing results obtained in the covariant and light-cone
approaches. Expressing our results in spinor helicity formalism we reproduce
the perhaps mysterious mismatch between the covariant approach and the light
cone approach in 4 dimensions. Our analysis also shows that there exists a
mismatch, in the 3-point gauge invariant amplitudes corresponding to cubic
self-interactions, between a scalar field $\phi$ and an antisymmetric rank-2
tensor field $A_{\mu\nu}$. Despite the well-known fact that in 4 dimensions
rank-2 anti-symmetric fields are dual to scalar fields in free theories, such
duality does not extend to interacting theories.
