Factorial preservation

Abstract.

Let A be a Dedekind domain with finite residue fields and with a finite unit group. Let S be an infinite subset of A and f be a polynomial with coefficients in the quotient field of A. We show that if the subsets S and f (S) have the same factorials (in Bhargava’s sense), then f is of degree 1. In particular, we answer Gilmer and Smith’s question [10] $({\text{when }}A = \mathbb{Z}){\text{:}}$ if S and f (S) are polynomially equivalent (in McQuillan’s sense), then f is of degree 1.