Values of the <i>F</i>‐pure threshold for homogeneous polynomials

We find a formula, in terms of n , d $n, d$ and p $p$ for the value F $F$ -pure threshold generic homogeneous polynomial degree $d$ $n$ variables over an algebraically closed field characteristic . also show that every all ⩾ 4 $d\geqslant 4$ not divisible by there always exist reduced polynomials k [ x y ] $k[x, y]$ whose is truncation base expansion 2 $\frac{2}{d}$ at some place; particular, f $f$ strictly less than deg ( ) $\frac{2}{\deg (f)}$ provide example to resolve, negatively, question proposed Hernandez, Núñez-Betancourt, Witt, Zhang, as whether list necessary restrictions they prove on forms are “minimal” ≫ 0 $p\gg 0$ On other hand, we evidence supporting refining their ideas, including identifying specific truncations thresholds computations conditions suffice (in characteristic) degrees up eight several situations.