Maximal period polynomials over \(\mathbb{Z}/(p^ d)\) (Q1201600)

This interesting paper contains a criterion for a polynomial \(f(x)\) with integer coefficients to have the maximal modulo \(p^ d\) period. The criterion involves the consideration of the discriminant and of the linear recurring sequences. The author invokes results of \textit{M. Ward} [Trans. Am. Math. Soc. 35, 600-628 (1933; Zbl 0007.24901)] and \textit{M. Hall} [Trans. Am. Math. Soc. 44, 196-218 (1938; Zbl 0019.19301)]. He also uses a criterion of primitiveness of polynomials over finite fields of \textit{Z. Dai} and \textit{M. Huang} [Chin. Sci. Bull. 36, 892-985 (1991; Zbl 0739.11055)]. The computation of the principal part of the discriminant is obtained. We notice that in the cases \(p=2,3,5\) or 7 the values of the principal part of the discriminant are explicitly given.