Northcott numbers for the house and the Weil height
Fabien Pazuki, Niclas Technau, Martin Widmer

TL;DR
This paper investigates the Northcott numbers related to the house and Weil height for subrings of algebraic numbers, revealing diverse behaviors and linking these properties to decidability and conjectures in number theory.
Contribution
It characterizes possible Northcott numbers for various heights and constructs fields demonstrating different behaviors, extending known results and addressing open questions.
Findings
Every t≥1 is the Northcott number for some ring of integers w.r.t. house.
Existence of fields with Northcott numbers in [t,2t] for Weil height.
Fields with different Northcott number behaviors for weighted Weil heights, including zero and infinity.
Abstract
For an algebraic number and , be the (logarithmic) Weil height, and be the -weighted (logarithmic) Weil height of . Let be a function on the algebraic numbers , and let . The Northcott number of , with respect to , is the infimum of all such that is infinite. This paper studies the set of Northcott numbers for subrings of for the house, the Weil height, and the -weighted Weil height. We show: (1) Every is the Northcott number of a ring of integers of a field w.r.t. the house. (2) For each there exists a field with Northcott…
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Taxonomy
TopicsAlgebraic Geometry and Number Theory · Meromorphic and Entire Functions · Analytic Number Theory Research
