Monic integer Chebyshev problem

Borwein, Peter; Pinner, Christopher; Pritsker, Igor E.

doi:10.1090/s0025-5718-03-01477-7

Cited by 17 publications

(27 citation statements)

References 16 publications

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“…In this paper we continue a study, recently initiated by Borwein, Pinner and Pritsker [2], of the monic integer transfinite diameter of a real interval. We write the normalized supremum on an interval I as…”

Section: Introduction and Resultsmentioning

confidence: 81%

“…For n = 1, t M ([0, 1]) = 1 2 , with Q(x) = 2x − 1 and P (x) = x(x − 1). This was the case too in [2,Section 5] in the proof of the Farey Interval Conjecture for small-denominator intervals.…”

Section: Definitions Conjectures and Further Resultsmentioning

confidence: 82%

“…It is well known that cap(I) = |I|/4 for an interval I of length |I|. Further, if |I| ≥ 4, then t Z (I) = t M (I) = cap(I) by [2], so that the challenge for evaluating t M (I), as for t Z (I), lies in intervals with |I| < 4. For these intervals we know from [2, Prop.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…Finally, part (f) follows from the fact that for |I| ≥ 4 we have t M (I) = t Z (I) = cap(I) = |I| 4 (see for instance [2]). …”

Section: Upper and Lower Bounds For L − (T) And L + (T) For Fixed Tmentioning

confidence: 99%

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The monic integer transfinite diameter

Hare

Smyth

2006

Math. Comp.

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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 M (I) is defined as the infimum of all such supremums. We show that if I has length 1, then t M (I) = 1 2 . We make three general conjectures relating to the value of t M (I) for intervals I of length less than 4. We also conjecture a value for t M ([0, b]) where 0 < b ≤ 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 M (I) on either side of t. Upper and lower bounds are given for these functions.We also consider the problem of determining t 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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Section: Introduction and Resultsmentioning

confidence: 81%

Section: Definitions Conjectures and Further Resultsmentioning

confidence: 82%

Section: Introduction and Resultsmentioning

confidence: 99%

“…Finally, part (f) follows from the fact that for |I| ≥ 4 we have t M (I) = t Z (I) = cap(I) = |I| 4 (see for instance [2]). …”

Section: Upper and Lower Bounds For L − (T) And L + (T) For Fixed Tmentioning

confidence: 99%

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The monic integer transfinite diameter

Hare

Smyth

2006

Math. Comp.

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“…In recent years a number of papers related to [24] were published. See [39], [86], and [138], for example, and the references therein.…”

Section: Corollary 93 Let a Be A Subarc Of The Unit Circle With Lenmentioning

confidence: 99%

Untitled

2010

St. Petersburg Math. J.

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Small polynomials with integer coefficients

Pritsker

2005

J. Anal. Math.

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We study the problem of minimizing the supremum norm, on a segment of the real line or on a compact set in the plane, by polynomials with integer coefficients. The extremal polynomials are naturally called integer Chebyshev polynomials. Their factors, zero distribution and asymptotics are the main subjects of this paper. In particular, we show that the integer Chebyshev polynomials for any infinite subset of the real line must have infinitely many distinct factors, which answers a question of Borwein and Erdélyi. Furthermore, it is proved that the accumulation set for their zeros must be of positive capacity in this case.We also find the first nontrivial examples of explicit integer Chebyshev constants for certain classes of lemniscates. Since it is rarely possible to obtain an exact value of integer Chebyshev constant, good estimates are of special importance.Introducing the methods of weighted potential theory, we generalize and improve the Hilbert-Fekete upper bound for integer Chebyshev constant. These methods also give bounds for the multiplicities of factors of integer Chebyshev polynomials, and lower bounds for integer Chebyshev constant. Moreover, all the mentioned bounds can be found numerically, by using various extremal point techniques, such as weighted Leja points algorithm. Applying our results in the classical case of the segment [0, 1], we improve the known bounds for the integer Chebyshev constant and the multiplicities of factors of the integer Chebyshev polynomials. 2000 Mathematics Subject Classification. Primary 11C08, 30C10; Secondary 31A05, 31A15.

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Monic integer Chebyshev problem

Cited by 17 publications

References 16 publications

The monic integer transfinite diameter

The monic integer transfinite diameter

Untitled

Small polynomials with integer coefficients

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