Periodical properties of multidimensional polynomial transformations of Galois – Eisenstein ring

The paper is concerned with $m$-dimensional polynomial transformations of Galois – Eisenstein ring $R$ (that is a finite commutative local ring of principal ideals). The maximum $L_m(R)$ cycle lengths of such polynomial transformations is estimated. Under condition $p > 2$, the constraint of the function $L_m$ on the class of Galois – Eisenstein rings having a power $q_n = p^{tn}$ and nilpotency index $n$ takes the maximum value on the Galois rings.