Polyfunctions over commutative rings

Abstract

A function [Formula: see text], where [Formula: see text] is a commutative ring with unit element, is called polyfunction if it admits a polynomial representative [Formula: see text]. Based on this notion, we introduce ring invariants which associate to [Formula: see text] the numbers [Formula: see text] and [Formula: see text], where [Formula: see text] is the subring generated by [Formula: see text]. For the ring [Formula: see text] the invariant [Formula: see text] coincides with the number theoretic Smarandache or Kempner function [Formula: see text]. If every function in a ring [Formula: see text] is a polyfunction, then [Formula: see text] is a finite field according to the Rédei–Szele theorem, and it holds that [Formula: see text]. However, the condition [Formula: see text] does not imply that every function [Formula: see text] is a polyfunction. We classify all finite commutative rings [Formula: see text] with unit element which satisfy [Formula: see text]. For infinite rings [Formula: see text], we obtain a bound on the cardinality of the subring [Formula: see text] and for [Formula: see text] in terms of [Formula: see text]. In particular we show that [Formula: see text]. We also give two new proofs for the Rédei–Szele theorem which are based on our results.