The monic integer transfinite diameter
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- by K. G. Hare and C. J. Smyth PDF
- Math. Comp. 75 (2006), 1997-2019 Request permission
Corrigendum: Math. Comp. 77 (2008), 1869-1869.
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.References
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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
- MR Author ID: 164180
- Email: c.smyth@ed.ac.uk
- 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.
- © Copyright 2006 American Mathematical Society
- Journal: Math. Comp. 75 (2006), 1997-2019
- MSC (2000): Primary 11C08; Secondary 30C10
- DOI: https://doi.org/10.1090/S0025-5718-06-01843-6
- MathSciNet review: 2240646