#### Vol. 6, No. 6, 2012

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Ideals of degree one contribute most of the height

### Aaron Levin and David McKinnon

Vol. 6 (2012), No. 6, 1223–1238
##### Abstract

Let $k$ be a number field, $f\left(x\right)\in k\left[x\right]$ a polynomial over $k$ with $f\left(0\right)\ne 0$, and ${\mathsc{O}}_{k,S}^{\ast }$ 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 $T\subset {\mathsc{O}}_{k,S}^{\ast }$, the prime ideals of ${\mathsc{O}}_{k}$ dividing $f\left(u\right)$, $u\in {\mathsc{O}}_{k,S}^{\ast }\setminus T$, mostly have degree one over $ℚ$; 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.

##### Keywords
heights, Diophantine approximation, polynomial values, elliptic curves, Vojta's conjecture
Primary: 11G50
Secondary: 11J25