Values of the F-pure threshold for homogeneous polynomials

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 >= 4 not divisible by p, there always exist reduced polynomials of degree d in k[x, y] 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 f whose F-pure threshold is strictly less than 2/deg(f). We provide an example to resolve, negatively, a question proposed by Hernandez, Nunez-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 p >> 0. 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.