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Mathematics of Computation

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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
Published electronically: June 16, 2006
Corrigendum: Math. Comp. 77 (2008), 1869
MathSciNet review: 2240646
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Abstract: We study the problem of finding nonconstant monic integer polynomials, normalized by their degree, with small supremum on an interval $ I$. The monic integer transfinite diameter $ t_{\mathrm{M}}(I)$ is defined as the infimum of all such supremums. We show that if $ I$ has length $ 1$, then $ t_{\mathrm{M}}(I) = \tfrac{1}{2}$.

We make three general conjectures relating to the value of $ t_{\mathrm{M}}(I)$ for intervals $ I$ of length less than $ 4$. We also conjecture a value for $ t_{\mathrm{M}}([0,b])$ where $ 0<b\le 1$. We give some partial results, as well as computational evidence, to support these conjectures.

We define functions $ L_{-}(t)$ and $ L_{+}(t)$, which measure properties of the lengths of intervals $ I$ with $ t_{\mathrm{M}}(I)$ on either side of $ t$. Upper and lower bounds are given for these functions.

We also consider the problem of determining $ t_{\mathrm{M}}(I)$ when $ I$ 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

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

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