2021/07/19 by Fabien Pazuki, Pazuki, Fabien, Niclas Technau +3 · 1 citation
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2107.09027
To appear in Bull. Lond. Math. Soc
arxiv created 2022/02/22 · arxiv updated 2022/02/23
For an algebraic number α and γ∈ ℝ, h(α) be the (logarithmic) Weil height, and hγ(α)=(degα)γh(α) be the γ-weighted (logarithmic) Weil height of α. Let f:ℚ→ [0,∞) be a function on the algebraic numbers ℚ, and let S⊂ ℚ. The Northcott number Nf(S) of S, with respect to f, is the infimum of all X≥ 0 such that \α∈ S; f(α)< X\ is infinite. This paper studies the set of Northcott numbers Nf(O) for subrings of ℚ for the house, the Weil height, and the γ-weighted Weil height. We show: (1) Every t≥ 1 is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each t≥ 0 there exists a field with Northcott number in [t,2t] w.r.t. the Weil height h(⋅). (3) For all 0≤ γ≤ 1 and γ'<γ there exists a field K with N_hγ'(K)=0 and Nhγ(K)=∞. For (1) we provide examples that satisfy an analogue of Julia Robinon's property (JR), examples that satisfy an analogue of Vidaux and Videla's isolation property, and examples that satisfy neither of those. Item (2) concerns a question raised by Vidaux and Videla due to its direct link with decidability theory via the Julia Robinson number. Item (3) is a strong generalisation of the known fact that there are fields that satisfy the Lehmer conjecture but which are not Bogomolov in the sense of Bombieri and Zannier.