The ring of polyfunctions over

Abstract

We study the ring of polyfunctions over The ring of polyfunctions over a commutative ring R with unit element is the ring of functions which admit a polynomial representative in the sense that for all This allows to define a ring invariant s which associates to a commutative ring R with unit element a value in The function s generalizes the number theoretic Smarandache function. For the ring we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number of polyfunctions over We also investigate algebraic properties of the ring of polyfunctions over In particular, we identify the additive subgroup of the ring and the ring structure itself. Moreover we derive formulas for the size of the ring of polyfunctions in several variables over and we compute the number of polyfunctions which are units of the ring.