On polynomial functions from \(\mathbb{Z}_ n\) to \(\mathbb{Z}_ m\)
Let \(\mathbb{Z}_n\) denote the ring of residues modulo \(n\). The author calls a function \(f: \mathbb{Z}_n\to \mathbb{Z}_m\) a polynomial function if there is a polynomial \(F\in \mathbb{Z} [x ]\) such that \(f(j) \equiv F(j) \pmod m\) for \(j=1, \dots, m-1\). (It should be observed that a different choice of representatives for \(\mathbb{Z}_n\) would lead to a different class.) With \(\lambda\) being the smallest integer such that \(m|\lambda!\) and \(\mu= \min(n, \lambda)\), he proves that these functions have a unique representation in the form \(\sum^{ \mu-1 }_{ k=0} a_k x^k \pmod m\) with \(0\leq a_k< m/ (m,k!)\). The total number of polynomial functions is determined and criteria are given for a function to be polynomial. It is shown that those pairs \((m,n)\) for which all functions are polynomial are exactly those for which \(n\) is not greater than the least prime factor of \(m\).
scientific article; zbMATH DE number 719455
scientific article