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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

   

 

Some comments on Garsia numbers


Authors: Kevin G. Hare and Maysum Panju
Journal: Math. Comp. 82 (2013), 1197-1221
MSC (2010): Primary 11K16, 11Y40
Published electronically: August 7, 2012
MathSciNet review: 3008855
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Abstract: A Garsia number is an algebraic integer of norm $ \pm 2$ such that all of the roots of its minimal polynomial are strictly greater than $ 1$ in absolute value. Little is known about the structure of the set of Garsia numbers. The only known limit point of positive real Garsia numbers was $ 1$ (given, for example, by the set of Garsia numbers $ 2^{1/n}$). Despite this, there was no known interval of [1,2] where the set of positive real Garsia numbers was known to be discrete and finite. The main results of this paper are:

  • An algorithm to find all (complex and real) Garsia numbers up to some fixed degree. This was performed up to degree $ 40$.
  • An algorithm to find all positive real Garsia numbers in an interval $ [c, d]$ with $ c > \sqrt {2}$.
  • There exist two isolated limit points of the positive real Garsia numbers greater than $ \sqrt {2}$. These are $ 1.618\cdots $ and $ 1.465\cdots $, the roots of $ z^2-z-1$ and $ z^3-z^2-1$, respectively. There are no other limit points greater than $ \sqrt {2}$.
  • There exist infinitely many limit points of the positive real Garsia numbers, including $ \lambda _{m,n}$, the positive real root of $ z^m -z^n - 1$, with $ m > n$.

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Additional Information

Kevin G. Hare
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo Ontario, Canada, N2L 3G1
Email: kghare@math.uwaterloo.ca

Maysum Panju
Affiliation: Department of Pure Mathematics, University of Waterloo, Waterloo Ontario, Canada, N2L 3G1
Email: mhpanju@rogers.com

DOI: http://dx.doi.org/10.1090/S0025-5718-2012-02636-6
Received by editor(s): September 7, 2011
Received by editor(s) in revised form: October 13, 2011
Published electronically: August 7, 2012
Additional Notes: The first author’s research was partially supported by NSERC
The second author’s research was supported by NSERC, the UW President’s Research Award, UW Undergraduate Research Internship, and the department of Pure Mathematics at the University of Waterloo
Computational support provided by CFI/OIT grant
Article copyright: © Copyright 2012 By the authors