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

TL;DR

This paper explores the structure and properties of the ring of polyfunctions over the finite ring ulus/nulus, introducing a new invariant, providing formulas for the number of polyfunctions, and analyzing algebraic features.

Contribution

It introduces a new invariant for rings, provides a unique polynomial representation for polyfunctions over ulus/nulus, and derives formulas for counting polyfunctions and units.

Findings

Derived a formula for the number of polyfunctions over ulus/nulus.

Provided a unique polynomial representation for polyfunctions.

Analyzed algebraic structure and identified units in the ring.

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.