Northcott numbers for the house and the Weil height
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Northcott numbers for the house and the Weil height. / Pazuki, Fabien; Technau, Niclas; Widmer, Martin.
I: Bulletin of the London Mathematical Society, Bind 54, Nr. 5, 2022, s. 1873-1897.Publikation: Bidrag til tidsskrift › Tidsskriftartikel › Forskning › fagfællebedømt
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TY - JOUR
T1 - Northcott numbers for the house and the Weil height
AU - Pazuki, Fabien
AU - Technau, Niclas
AU - Widmer, Martin
N1 - Publisher Copyright: © 2022 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
PY - 2022
Y1 - 2022
N2 - For an algebraic number (Formula presented.) and (Formula presented.), let (Formula presented.) be the house, (Formula presented.) be the (logarithmic) Weil height, and (Formula presented.) be the (Formula presented.) -weighted (logarithmic) Weil height of (Formula presented.). Let (Formula presented.) be a function on the algebraic numbers (Formula presented.), and let (Formula presented.). The Northcott number (Formula presented.) of (Formula presented.), with respect to (Formula presented.), is the infimum of all (Formula presented.) such that (Formula presented.) is infinite. This paper studies the set of Northcott numbers (Formula presented.) for subrings of (Formula presented.) for the house, the Weil height, and the (Formula presented.) -weighted Weil height. We show: (1)Every (Formula presented.) is the Northcott number of a ring of integers of a field w.r.t. the house (Formula presented.). (2)For each (Formula presented.), there exists a field with Northcott number in (Formula presented.) w.r.t. the Weil height (Formula presented.). (3)For all (Formula presented.) and (Formula presented.), there exists a field (Formula presented.) with (Formula presented.) and (Formula presented.). 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.
AB - For an algebraic number (Formula presented.) and (Formula presented.), let (Formula presented.) be the house, (Formula presented.) be the (logarithmic) Weil height, and (Formula presented.) be the (Formula presented.) -weighted (logarithmic) Weil height of (Formula presented.). Let (Formula presented.) be a function on the algebraic numbers (Formula presented.), and let (Formula presented.). The Northcott number (Formula presented.) of (Formula presented.), with respect to (Formula presented.), is the infimum of all (Formula presented.) such that (Formula presented.) is infinite. This paper studies the set of Northcott numbers (Formula presented.) for subrings of (Formula presented.) for the house, the Weil height, and the (Formula presented.) -weighted Weil height. We show: (1)Every (Formula presented.) is the Northcott number of a ring of integers of a field w.r.t. the house (Formula presented.). (2)For each (Formula presented.), there exists a field with Northcott number in (Formula presented.) w.r.t. the Weil height (Formula presented.). (3)For all (Formula presented.) and (Formula presented.), there exists a field (Formula presented.) with (Formula presented.) and (Formula presented.). 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.
UR - http://www.scopus.com/inward/record.url?scp=85129761752&partnerID=8YFLogxK
U2 - 10.1112/blms.12662
DO - 10.1112/blms.12662
M3 - Journal article
AN - SCOPUS:85129761752
VL - 54
SP - 1873
EP - 1897
JO - Bulletin of the London Mathematical Society
JF - Bulletin of the London Mathematical Society
SN - 0024-6093
IS - 5
ER -
ID: 307095398