The Difference Between Permutation Polynomials Over Finite Fields
Stephen D. Cohen, Gary L. Mullen, Peter J.-S. Shiue · Proceedings of the American Mathematical Society · 1995
Recently S. D. Cohen resolved a conjecture of Chowla and Zassenhaus (1968) in the affirmative by showing that, if $f(x)$ and $g(x)$ are integral polynomials of degree $n \geq 2$ and p is a prime exceeding ${({n^2} - 3n + 4)^2}$ for which f and g are both permutation polynomials of the finite field ${F_p}$, then their difference $h = f - g$ cannot be such that $h(x) = cx$ for some integer c not divisible by p. In this note we provide a significant generalization by proving that, if h is not a constant in ${F_p}$ and t is the degree of h, then $t \geq 3n/5$ and, provided $t \leq n - 3$, t and n are not relatively prime. In a sense this measures the isolation of permutation polynomials of the same degree over large finite prime fields.