Enumerating local permutation polynomials over residue class rings
For a prime \(p\) and a positive integer \(n\), let \(f(x,y)\) denote a polynomial in two variables over \(\mathbb{Z}/p^n \mathbb{Z}\). Then \(f\) is called a local permutation polynomial (LPP) modulo \(p^n\) if for each \(a\) in \(\mathbb{Z}/p^n \mathbb{Z}\) the polynomials (in one variable) \(f(a,y)\) and \(f(x,a)\) are permutation polynomials, i.e., the respective induced maps from \(\mathbb{Z}/p^n \mathbb{Z}\) to itself are bijective. The number of LPPs and the degree of LPPs are investigated. We sketch one of the two main results of the paper (the other one deals with the degree). For a fixed prime \(p\), let \(\ell_n\) denote the number of LPPs modulo \(p^n\) (identifying those giving rise to the same map). Then \(\ell_2 = p^{p^2}(p-1)^{2p^2}\ell_1\) and for \(n>2\) one has \(\ell_n = p^{\omega_n} \ell_{n-1}\) with \(\omega_n\) some constant whose definition is too long to be recalled here.
scientific article; zbMATH DE number 5640589