Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Quantitative height bounds under splitting conditions


Authors: Paul A. Fili and Lukas Pottmeyer
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11G50, 11R06, 37P30, 31A15
DOI: https://doi.org/10.1090/tran/7656
Published electronically: June 21, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In an earlier work, the first author and Petsche used potential theoretic techniques to establish a lower bound for the height of algebraic numbers that satisfy splitting conditions such as being totally real or $ p$-adic, improving on earlier work of Bombieri and Zannier in the totally $ p$-adic case. These bounds applied as the degree of the algebraic number over the rationals tended towards infinity. In this paper, we use discrete energy approximation techniques on the Berkovich projective line to make the dependence on the degree in these bounds explicit, and we establish lower bounds for algebraic numbers which depend only on local properties of the numbers.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11G50, 11R06, 37P30, 31A15

Retrieve articles in all journals with MSC (2010): 11G50, 11R06, 37P30, 31A15


Additional Information

Paul A. Fili
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma 74078
Email: fili@post.harvard.edu

Lukas Pottmeyer
Affiliation: Fakultät für Mathematik, Universität Duisburg-Essen, D-45117 Essen, Germany
Email: lukas.pottmeyer@uni-due.de

DOI: https://doi.org/10.1090/tran/7656
Keywords: Weil height, totally real, totally $p$-adic, splitting conditions
Received by editor(s): August 6, 2015
Received by editor(s) in revised form: August 3, 2017, and June 20, 2018
Published electronically: June 21, 2019
Additional Notes: The second author was supported by the DFG-Projekt Heights and unlikely intersections HA 6828/1-1.
Article copyright: © Copyright 2019 American Mathematical Society