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 Fq[t] over a finite field Fq and then define the Smarandache function of a non-zero polynomial f∈Fq[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 Erdős, which implies that for almost all polynomials f, S(f)=td, where d is the maximal degree of the irreducible factors of f.