Polynomial analogue of the Smarandache function

Abstract

In the integer case, the Smarandache function of a positive integer $n$ is defined to be the smallest positive integer $k$ such that $n$ divides the factorial $k!$. In this paper, we first define a natural order for polynomials in $\mathbb{F}_q[t]$ over a finite field $\mathbb{F}_q$ and then define the Smarandache function of a non-zero polynomial $f \in \mathbb{F}_q[t]$, denoted by $S(f)$, to be the smallest polynomial $g$ such that $f$ divides the Carlitz factorial of $g$. In particular, we establish an analogue of a problem of Erd{\H o}s, which implies that for almost all polynomials $f$, $S(f)=t^d$, where $d$ is the maximal degree of the irreducible factors of $f$.