  We find a formula, in terms of n, d and p, for the value of the F-pure
threshold for the generic homogeneous polynomial of degree d in n variables
over an algebraically closed field of characteristic p. We also show that, in
every characteristic p and for all d (greater than 3) not divisible by p, there
always exist reduced polynomials of degree d whose F-pure threshold is a
truncation of the base p expansion of 2/d at some place; in particular, there
always exist reduced polynomials whose F-pure threshold is strictly less than
2/d. We provide an example to resolve, negatively, a question proposed by
Hernandez, N\'u\~nez-Betancourt, Witt and Zhang, as to whether a list of
necessary restrictions they prove on the F-pure threshold of reduced forms are
"minimal" for large p. On the other hand, we also provide evidence supporting
and refining their ideas, including identifying specific truncations of the
base p expansion of 2/d that are always F-pure thresholds for reduced forms of
degree d, and computations that show their conditions suffice (in every
characteristic) for degrees up to eight and several other situations. Finally,
we point out a lower bound on the F-pure threshold of a reduced form in terms
of its degree and the characteristic p.
