Polynomial analogue of the Smarandache function

TL;DR

This paper introduces a polynomial analogue of the Smarandache function over finite fields, establishing properties and an Erdős-type problem related to divisibility and factorials of polynomials.

Contribution

It defines a natural order for polynomials over finite fields and introduces the Smarandache function for polynomials, extending classical concepts to algebraic function fields.

Findings

For almost all polynomials, S(f) equals t^d, where d is the maximum degree of irreducible factors.

Established an analogue of Erdős's problem in the polynomial setting.

Provided foundational definitions and properties for polynomial Smarandache functions.

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 $f$.