Geodesic
Polynomials of maximal period over primary residue rings
Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 549-551.

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The maximality criterion for the period of a polynomial over primary residue ring is proved. This criterion generalize the results of the paper [1], where the case of polynomials over $\mathbb Z_{2^n}$ was considered, to the case of arbitrary primary ring $\mathbb Z_{p^n}$. The criterion is based on the concept of “marked polynomial” introduced in [1] and allows to verify the maximality of the period of a polynomial using only its coefficients. Some sufficient conditions of maximality of the period of a polynomial over $\mathbb Z_{p^n}$ are given. 
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A. S. Kuz'min. Polynomials of maximal period over primary residue rings. Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 549-551. http://geodesic.mathdoc.fr/item/FPM_1995_1_2_a17/

[1] Nechaev A. A., “Tsiklovye tipy lineinykh podstanovok nad konechnymi kommutativnymi koltsami”, Mat. sb., 184:3 (1993), 21–56 | MR | Zbl

[2] Ward M., “The arithmetical theory of linear recurring series”, Trans. Amer. Math. Soc., 35 (1933) | MR