Mason's theorem and syzygy gaps

From Mason's theorem on rational function fields (the progenitor of the abc-conjecture) we immediately derive upper bounds for “syzygy gaps” (Theorems 3, 8, 11, and Corollary 9). These in turn quickly give:(1)The author's conjecture Z(l), used in the study of Hilbert–Kunz series.(2)A lower bound for certain “F-pure thresholds.”(3)Han's explicit description of the 3-dimensional p-fractal attached to x, y and x+y.(4)Some apparently simple degree estimates, observed in the past but unproved until now (Corollary 6 and Theorem 10).