The Ring of Polyfunctions over $\mathbb{Z}/n\mathbb{Z}$

Abstract

We study the ring of polyfunctions over $\mathbb{Z}/n\mathbb{Z}$. The ring of polyfunctions over a commutative ring $R$ with unit element is the ring of functions $f:R\to R$ which admit a polynomial representative $p\in R[x]$ in the sense that $f(x)=p(x)$ for all $x\in R$. This allows to define a ring invariant $s$ which associates to a commutative ring $R$ with unit element a value in $\mathbb{N}\cup \{\infty \}$. The function $s$ generalizes the number theoretic Smarandache function. For the ring $R=\mathbb{Z}/n\mathbb{Z}$ we provide a unique representation of polynomials which vanish as a function. This yields a new formula for the number $\Psi (n)$ of polyfunctions over $\mathbb{Z}/n\mathbb{Z}$. We also investigate algebraic properties of the ring of polyfunctions over $\mathbb{Z}/n\mathbb{Z}$. 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 $\mathbb{Z}/n\mathbb{Z}$, and we compute the number of polyfunctions which are units of the ring.

26 pages. Communications in Algebra, 2022

References in corpus (2)

• On the group of unit-valued polynomial functions
• Polyfunctions over Commutative Rings