The monic integer transfinite diameter

Authors:
K. G. Hare and C. J. Smyth

Journal:
Math. Comp. **75** (2006), 1997-2019

MSC (2000):
Primary 11C08; Secondary 30C10

DOI:
https://doi.org/10.1090/S0025-5718-06-01843-6

Published electronically:
June 16, 2006

Corrigendum:
Math. Comp. 77 (2008), 1869

MathSciNet review:
2240646

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval . The monic integer transfinite diameter is defined as the infimum of all such supremums. We show that if has length , then .

We make three general conjectures relating to the value of for intervals of length less than . We also conjecture a value for where . We give some partial results, as well as computational evidence, to support these conjectures.

We define functions and , which measure properties of the lengths of intervals with on either side of . Upper and lower bounds are given for these functions.

We also consider the problem of determining when is a Farey interval. We prove that a conjecture of Borwein, Pinner and Pritsker concerning this value is true for an infinite family of Farey intervals.

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Additional Information

**K. G. Hare**

Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada N2L 3G1

Email:
kghare@math.uwaterloo.ca

**C. J. Smyth**

Affiliation:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom

Email:
c.smyth@ed.ac.uk

DOI:
https://doi.org/10.1090/S0025-5718-06-01843-6

Keywords:
Chebyshev polynomials,
monic integer transfinite diameter

Received by editor(s):
April 21, 2005

Received by editor(s) in revised form:
June 20, 2005

Published electronically:
June 16, 2006

Additional Notes:
Research of the first author was supported in part by NSERC of Canada and a Seggie Brown Fellowship, University of Edinburgh.

Article copyright:
© Copyright 2006
American Mathematical Society