On polynomials with largest coefficient sums

Rack, Heinz-Joachim

doi:10.1016/0021-9045(89)90124-x

Cited by 7 publications

(10 citation statements)

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“…Such polynomials of degree 2 ≤ n ≤ 4 and represented in a power form are scattered in the literature, see [5], [7], [10], [22, p. 156], [27], [28], [29], [34] and [39, p. 98] (the latter with respect to [0, 1]). They can be expressed (possibly after some rearrangements) analytically as…”

Section: Explicit Analytical One-parameter Power Form Representation ...mentioning

confidence: 99%

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An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials

Rack¹,

Vajda²

2019

Preprint

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The problem to determine an explicit one-parameter power form representation of the proper Zolotarev polynomials of degree n and with uniform norm 1 on [−1, 1] can be traced back to P. L. Chebyshev. It turned out to be complicated, even for small values of n. Such a representation was known to A. A. Markov (1889) for n = 2 and n = 3. But already for n = 4 it seems that nobody really believed that an explicit form can be found. As a matter of fact it was, by V. A. Markov in 1892, as A. Shadrin put it in 2004. About 125 years passed before an explicit form for the next higher degree, n = 5, was found, by G. Grasegger and N. Th. Vo (2017). In this paper we settle the case n = 6.

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Section: Explicit Analytical One-parameter Power Form Representation ...mentioning

confidence: 99%

“…They find application (after rescaling) e.g. in the proof of the Markov inequality [17], [19] and Landau-Kolmogorov inequality [35], in the proof of Schur's Markov-type problem [8], [28], [29] and in the problem of maximizing linear coefficient functionals [27].…”

Section: Introduction and Historical Remarksmentioning

confidence: 99%

An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials

Rack¹,

Vajda²

2019

Preprint

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“…In ( [12], p. 357) we have provided explicit expressions for the parameterized coefficients of an arbitrary fourth-degree hard-core Zolotarev polynomial on I. But since the assumption was made there that it attains the value 1 at x = −1, we prefer to consider here the negative form of that polynomial in order to be compliant with [3].…”

Section: Alternative Deductions Of An Explicit Extremal Hard-core Zolmentioning

confidence: 99%

“…In Section 4 we have alternatively deduced the Schur polynomial P * 4 from the explicit power form Z 4,t (x) = ... as given, up to the sign, in ( [12], p. 357). P * 4 can likewise be deduced from the explicit power form Z 4 (x, t) = ... as given in ( [15], p. 242), however instead of Z (2) 4,t (1) = 0 (see (4.4)) one has then to set Z (2) 4 (−1, t) = 0.…”

Section: A Generalized Schur Problem and Its Solution For The Quarticmentioning

confidence: 99%

“…Our first result centers around the proof in [3] for the initial case n = 4 and is three-fold: (i) the numerical value for (1) in ([3], (7.9)) is not correct, but sufficiently precise; (ii) from preliminary work in [3] can in fact be deduced a closed analytic expression for (1) if n = 4, allowing numerical evaluation to any precision; (iii) even the explicit power form representation of an extremal Z4,t = Z4,t * can be deduced from [3], thus providing an exemplification of Schur's problem that seems to be novel. Additionally, we cross-check its validity twice: firstly by deriving Z4,t * conversely from a general formula for Z4,t that we have given in [12], and secondly by applying Theorem 5.10 in [11]. We then turn to a generalized solution of Schur's problem, to k -th derivatives, by Shadrin [16].…”

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The first Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of Schur

Rack

2017

Stud. Univ. Babes-Bolyai Math.

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Abstract. Schur's [14] Markov-type extremal problem asks to find the maximum (1) sup −1≤ξ≤1 sup Pn∈B n,ξ,2 |P(1) n (ξ)|, where B n,ξ,2 = {Pn ∈ Bn : P (2) n (ξ) = 0} ⊂ Bn = {Pn : |Pn(x)| ≤ 1 for |x| ≤ 1} and Pn is an algebraic polynomial of degree ≤ n. Erdös and Szegö [3] found that for n ≥ 4 this maximum is attained if ξ = ±1 and Pn ∈ B n,ξ,2 is a (unspecified) member of the 1-parameter family of hard-core Zolotarev polynomials Zn,t. Our first result centers around the proof in [3] for the initial case n = 4 and is three-fold: (i) the numerical value for (1) in ([3], (7.9)) is not correct, but sufficiently precise; (ii) from preliminary work in [3] can in fact be deduced a closed analytic expression for (1) if n = 4, allowing numerical evaluation to any precision; (iii) even the explicit power form representation of an extremal Z4,t = Z4,t * can be deduced from [3], thus providing an exemplification of Schur's problem that seems to be novel. Additionally, we cross-check its validity twice: firstly by deriving Z4,t * conversely from a general formula for Z4,t that we have given in [12], and secondly by applying Theorem 5.10 in [11]. We then turn to a generalized solution of Schur's problem, to k -th derivatives, by Shadrin [16]. Again we provide in explicit form the corresponding maximum as well as an extremizer polynomial for the first non-trivial degree n = 4. Finally, we contribute to the fuller description of Z4,t by providing its critical points in explicit form.Mathematics Subject Classification (2010): 26C05, 26D10, 41A10, 41A17, 41A29, 41A44, 41A50.

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An Algebraic-Geometric Method for Computing Zolotarev Polynomials

Grasegger

2017

Proceedings of the 2017 ACM International Symposium on Symbolic and Algebraic Computation

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On polynomials with largest coefficient sums

Cited by 7 publications

References 7 publications

An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials

An explicit univariate and radical parametrization of the sextic proper Zolotarev polynomials

The first Zolotarev case in the Erdös-Szegö solution to a Markov-type extremal problem of Schur

An Algebraic-Geometric Method for Computing Zolotarev Polynomials

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