zbMATH — the first resource for mathematics

Permutation polynomials in several variables over residue class rings. (English) Zbl 0644.13003
There are two ways in generalizing the concept of permutation polynomial functions (PPFs) over a commutative ring R with 1 to multiplaced functions, namely \(k\)-ary PPFs and strict \(k\)-ary PPFs. These can be defined as polynomial functions \(f: R^ k\to R\) for which k-ary functions (resp. polynomial functions) \(f_ 2,...,f_ k\) over \(R\) exist, such that the \(k\)-tuple \((f,f_ 2,...,f_ k)\) represents a permutation of \(R^ k\). It is shown that if R is isomorphic to the direct product \(\times^{n}_{i=1}{\mathbb{Z}}_{p_ i^{e_ i}}\) then these two concepts coincide if and only if all \(e_ i=1\).
Reviewer: G.Kowol

13B25 Polynomials over commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
05A05 Permutations, words, matrices