Equidistribution of points via energy

Pritsker, Igor E.

doi:10.1007/s11512-010-0124-2

Cited by 19 publications

(17 citation statements)

References 35 publications

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“…The theorem of Erdős and Turán has been extended by several authors (see, for instance [3,2,19,20,21]). In particular, Mignotte showed that one can use the weaker norm M + in estimates like (1.3), see [19] and [2].…”

Section: Annular Discrepancies and Norms On Tmentioning

confidence: 99%

Expected discrepancy for zeros of random algebraic polynomials

Pritsker

Sola

2014

Proc. Amer. Math. Soc.

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Abstract. We study asymptotic clustering of zeros of random polynomials, and show that the expected discrepancy of roots of a polynomial of degree n, with not necessarily independent coefficients, decays like √ log n/n. Our proofs rely on discrepancy results for deterministic polynomials, and order statistics of a random variable. We also consider the expected number of zeros lying in certain subsets of the plane, such as circles centered on the unit circumference, and polygons inscribed in the unit circumference.

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Section: Annular Discrepancies and Norms On Tmentioning

confidence: 99%

Expected discrepancy for zeros of random algebraic polynomials

Pritsker

Sola

2014

Proc. Amer. Math. Soc.

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“…His work originated several important directions in analysis and number theory. Certain number theoretic aspects of Schur's paper are discussed in detail in [18], while [19] emphasizes its analytic side as generalized by Fekete [5] and Szegő [25]. We review and develop some of these recent results on Schur's problems from [21].…”

Section: Integer Chebyshev Problemmentioning

confidence: 99%

“…and note that each Q n has simple zeros {z j,n } n j=1 at the roots of unity, and integer coefficients. Using number theoretic arguments, we show in Remark 2.8 of [19] that the degree of Q n is…”

Section: Means Of Zerosmentioning

confidence: 99%

Polynomials with integer coefficients and their zeros

Pritsker

2012

J Math Sci

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Abstract. We study several related problems on polynomials with integer coefficients. This includes the integer Chebyshev problem, and the Schur problems on means of algebraic numbers. We also discuss interesting applications to the approximation by polynomials with integer coefficients, and to the growth of coefficients for polynomials with roots located in prescribed sets. The distribution of zeros for polynomials with integer coefficients plays an important role in all of these problems.Keywords. Polynomials, distribution of zeros, algebraic numbers. Integer Chebyshev problemLet C n and Z n be the classes of algebraic polynomials of degree at most n, respectively with complex and with integer coefficients. Define the uniform norm on a compact set E ⊂ C byThe problem of minimizing the uniform norm on E by monic polynomials from C n is well known as the Chebyshev problem (see [8,20,30], etc.) In the classical case E = [−1, 1], the explicit solution of this problem is given by the monic Chebyshev polynomial T n (x) := 2 1−n cos(n arccos x) of degree n ∈ N. Using a change of variable, we extend this to an arbitrary interval [a, b] ⊂ R, so thatis a unique monic polynomial with real coefficients and the smallest uniform norm on [a, b] among all monic polynomials of exact degree n from C n . It is immediate thatThe Chebyshev problem and the Chebyshev polynomials penetrated far beyond the original area of application in analysis, and these ideas remain of fundamental importance, cf. [20]. Many connections and generalizations were found in various areas of approximation theory, complex analysis, special

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“…It converges monotonically to the transfinite diameter of S 2 (in the logarithmic case), introduced in [PoSz31], which equals the logarithmic capacity exp(−W log ) of S 2 ; see [Pri11] for a recent account.…”

Section: The Logarithmic Casementioning

confidence: 99%