  Let $(X,L)$ be any Fano manifold polarized by a positive multiple of its
fundamental divisor $H$. The polynomial defining the Hilbert curve of $(X,L)$
boils down to being the Hilbert polynomial of $(X,H)$, hence it is totally
reducible over $\mathbb C$; moreover, some of the linear factors appearing in
the factorization have rational coefficients, e.g. if $X$ has index $\geq 2$.
It is natural to ask when the same happens for all linear factors. Here the
total reducibility over $\mathbb Q$ of the Hilbert polynomial is investigated
for three special kinds of Fano manifolds: Fano manifolds of large index, toric
Fano manifolds of low dimension, and Fano bundles of low coindex.
