## The integer Chebyshev problem

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- by Peter Borwein and Tamás Erdélyi PDF
- Math. Comp.
**65**(1996), 661-681 Request permission

## Abstract:

We are concerned with the problem of minimizing the supremum norm on an interval of a nonzero polynomial of degree at most $n$ with integer coefficients. This is an old and hard problem that cannot be exactly solved in any nontrivial cases. We examine the case of the interval $[0,1]$ in most detail. Here we improve the known bounds a small but interesting amount. This allows us to garner further information about the structure of such minimal polynomials and their factors. This is primarily a (substantial) computational exercise. We also examine some of the structure of such minimal “integer Chebyshev” polynomials, showing for example that on small intevals $[0, \delta ]$ and for small degrees $d$, $x^{d}$ achieves the minimal norm. There is a natural conjecture, due to the Chudnovskys and others, as to what the “integer transfinite diameter” of $[0,1]$ should be. We show that this conjecture is false. The problem is then related to a trace problem for totally positive algebraic integers due to Schur and Siegel. Several open problems are raised.## References

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## Additional Information

**Peter Borwein**- Affiliation: Department of Mathematics and Statistics, Simon Fraser University, Burnaby, B.C., Canada V5A 1S6
- Email: pborwein@cecm.sfu.ca
**Tamás Erdélyi**- Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
- Email: terdelyi@math.tamu.edu
- Received by editor(s): April 25, 1994
- Received by editor(s) in revised form: September 5, 1994, and February 12, 1995
- Additional Notes: Research supported in part by NSERC of Canada.
- © Copyright 1996 American Mathematical Society
- Journal: Math. Comp.
**65**(1996), 661-681 - MSC (1991): Primary 11J54, 11B83
- DOI: https://doi.org/10.1090/S0025-5718-96-00702-8
- MathSciNet review: 1333305