  Tree-level dynamical stability of scalar field potentials in renormalizable
theories can in principle be expressed in terms of positivity conditions on
quartic polynomial structures. However, these conditions cannot always be cast
in a fully analytical resolved form, involving only the couplings and being
valid for all field directions. In this paper we consider such forms in three
physically motivated models involving $SU(2)$ triplet scalar fields: the
Type-II seesaw model, the Georgi-Machacek model, and a generalized two-triplet
model. A detailed analysis of the latter model allows to establish the full set
of necessary and sufficient boundedness from below conditions. These can serve
as a guide, together with unitarity and vacuum structure constraints, for
consistent phenomenological (tree-level) studies. They also provide a seed for
improved loop-level conditions, and encompass in particular the leading ones
for the more specific Georgi-Machacek case. Incidentally, we present complete
proofs of various properties and also derive general positivity conditions on
quartic polynomials that are equivalent but much simpler than the ones used in
the literature.
