On the power polynomial \(x^ d\) over Galois rings
For a prime \(p\) let \(GF(p^ n,m)\) denote the Galois ring of order \(p^{mn}\) which can be obtained as a Galois extension of \(\mathbb{Z}_{p^ n}\). The cardinality of the sets \(\{P_ d(x):x\in GR(p^ n,m)\}\) and of the preimage of \(P_ d(x_ 0)\) are determined where \(P_ d(x)=x^ d\). The cases of \(p\) odd and even are treated separately and the enumeration of the preimage cardinality is a function of \(i\) where \(x_ 0=p^ i A\) for \(A\) a unit of \(GF(p^ n,m)\).
scientific article; zbMATH DE number 168033