Polynomials with integer coefficients and their zeros

Pritsker, Igor E.

doi:10.1007/s10958-012-0842-z

Cited by 2 publications

(2 citation statements)

References 24 publications

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“…On the other hand, Theorem 2.7 of [17] suggests the conjecture ℓ m = ℓ m , m ∈ N. We give more evidence in support of this conjecture below.…”

Section: Symmetric Means Of Algebraic Numbersmentioning

confidence: 73%

Asymptotic distribution and symmetric means of algebraic numbers

Pritsker

2015

Acta Arith.

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Schur introduced the problem on the smallest limit point for the arithmetic means of totally positive conjugate algebraic integers. This area was developed further by Siegel, Smyth and others. We consider several generalizations of the problem that include questions on the smallest limit points of symmetric means. The key tool used in the study is the asymptotic distribution of algebraic numbers understood via the weak* limits of their counting measures. We establish interesting properties of the limiting measures, and find the smallest limit points of symmetric means for totally positive algebraic numbers of small height. Schur's problems on means of algebraic numbersSchur [21] considered several problems for various classes of polynomials with integers coefficients and zeros restricted to certain sets. For convenience, we introduce the following notation. Let E be a subset of the complex plane C. Consider the set of polynomials Z n (E) with integer coefficients of the exact degree n and all zeros in E. We denote the subset of Z n (E) with simple zeros by Z s n (E). Given M > 0, we write P n = a n z n + . . . ∈ Z s n (E, M) if |a n | ≤ M and P n ∈ Z s n (E) (respectively P n ∈ Z n (E, M) if |a n | ≤ M and

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“…On the other hand, Theorem 2.7 of [17] suggests the conjecture ℓ m = ℓ m , m ∈ N. We give more evidence in support of this conjecture below.…”

Section: Symmetric Means Of Algebraic Numbersmentioning

confidence: 73%

Asymptotic distribution and symmetric means of algebraic numbers

Pritsker

2015

Acta Arith.

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“…The multivariate case has been looked at in [3,17]. Applications of this to the leading coefficient in Schur's problem were studied in [21].…”

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Applications of Integer Semi-Infinite Programing to the Integer Chebyshev Problem

Hare

Hodges

2019

Experimental Mathematics

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We consider the integer Chebyshev problem, that of minimizing the supremum norm over polynomials with integer coefficients on the interval [0, 1]. We implement algorithms from semi-infinite programming and a branch and bound algorithm to improve on previous methods for finding integer Chebyshev polynomials of degree n. Using our new method, we found 16 new integer Chebyshev polynomials of degrees in the range 147 to 244.Here this constant is explicitly computable to an arbitrary number of digits. See [11,18] for details. In [5] it was shown that the lower bound coming from this infinite family is in fact not best possible. That is, there exists an ǫ > 0 such that t Z ([0, 1]) ≥ 0.4207263 · · · + ǫ. At the time no non-trivial lower bound for ǫ was determined. Pritsker showed in [19], by means of weighted potential theory, that t Z ([0, 1]) ≥ 0.4213. Generalizations of these Gorshkov-Wirsing polynomails were considered in [15].Given the submultiplicative nature of t Z,n (I) we have t Z (I) ≤ t Z,n (I) for all n. This gives a simple method to find an upper bound for t Z (I); find large degree polynomials with small supremum norm. In [5] a set of 9 polynomials p i (x) and exponents a i were found such that the resulting polynomial P (x) = p 1 (x) a1 . . . p 9 (x) a9 had small supremum norm. This was used to show that t Z ([0, 1]) ≤

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Polynomials with integer coefficients and their zeros

Cited by 2 publications

References 24 publications

Asymptotic distribution and symmetric means of algebraic numbers

Asymptotic distribution and symmetric means of algebraic numbers

Applications of Integer Semi-Infinite Programing to the Integer Chebyshev Problem

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