Northcott numbers for the house and the Weil height

Abstract

For an algebraic number α $\alpha$ and γ ∈ R $\gamma \in \mathbb {R}$ , let be the house, h ( α ) $h(\alpha )$ be the (logarithmic) Weil height, and h γ ( α ) = ( deg α ) γ h ( α ) $h_\gamma (\alpha )=(\deg \alpha )^\gamma h(\alpha )$ be the γ $\gamma$ -weighted (logarithmic) Weil height of α $\alpha$ . Let f : Q ¯ → [ 0 , ∞ ) $f:\overline{\mathbb {Q}}\rightarrow [0,\infty )$ be a function on the algebraic numbers Q ¯ $\overline{\mathbb {Q}}$ , and let S ⊂ Q ¯ $S\subset \overline{\mathbb {Q}}$ . The Northcott number N f ( S ) $\mathcal {N}_f(S)$ of S $S$ , with respect to f $f$ , is the infimum of all X ⩾ 0 $X\geqslant 0$ such that { α ∈ S ; f ( α ) < X } $\lbrace \alpha \in S; f(\alpha )< X\rbrace$ is infinite. This paper studies the set of Northcott numbers N f ( O ) $\mathcal {N}_f({\mathcal {O}})$ for subrings of Q ¯ $\overline{\mathbb {Q}}$ for the house, the Weil height, and the γ $\gamma$ -weighted Weil height. We show: (1) Every t ⩾ 1 $t\geqslant 1$ is the Northcott number of a ring of integers of a field w.r.t. the house . (2) For each t ⩾ 0 $t\geqslant 0$ , there exists a field with Northcott number in [ t , 2 t ] $ [t,2t]$ w.r.t. the Weil height h ( · ) $h(\cdot )$ . (3) For all 0 ⩽ γ ⩽ 1 $0\leqslant \gamma \leqslant 1$ and γ ′ < γ $\gamma ^{\prime }<\gamma$ , there exists a field K $K$ with N h γ ′ ( K ) = 0 $\mathcal {N}_{h_{\gamma ^{\prime }}}(K)=0$ and N h γ ( K ) = ∞ $\mathcal {N}_{h_\gamma }(K)=\infty$ . For (1) we provide examples that satisfy an analogue of Julia Robinson'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.