Ideals of degree one contribute most of the height

Let $k$ be a number field, $f(x)\in k[x]$ a polynomial over $k$ with $f(0)\mathrm{\ne }0$, and ${\text{O}}_{k,S}^{\ast }$ the group of $S$-units of $k$, where $S$ is an appropriate finite set of places of $k$. In this note, we prove that outside of some natural exceptional set $T\subset {\text{O}}_{k,S}^{\ast }$, the prime ideals of ${\text{O}}_k$ dividing $f(u)$, $u\in {\text{O}}_{k,S}^{\ast }\setminus T$, mostly have degree one over $\text{Q}$; that is, the corresponding residue fields have degree one over the prime field. We also formulate a conjectural analogue of this result for rational points on an elliptic curve over a number field, and deduce our conjecture from Vojta’s Conjecture. We prove this conjectural analogue in certain cases when the elliptic curve has complex multiplication.