Improved Lower Bound for the Mahler Measure of the Fekete Polynomials

Erdélyi, Tamás

doi:10.1007/s00365-017-9398-y

Cited by 3 publications

(2 citation statements)

References 30 publications

(27 reference statements)

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“…Since

f_{p}

has constant coefficient 0, it is not a Littlewood polynomial, but

g_{p}

defined by

g_{p} false(z false) : = f_{p} false(z false) / z

is a Littlewood polynomial of degree

p - 2

. Fekete polynomials are examined in detail in . In the authors examined the maximal size of the Mahler measure and the

L_{p}

norms of sums of n monomials on the unit circle as well as on subarcs of the unit circles.…”

Section: Introduction and Notationmentioning

confidence: 99%

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On the Oscillation of the Modulus of the Rudin–shapiro Polynomials on the Unit Circle

Erdélyi

2019

Mathematika

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In signal processing the Rudin–Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar and communication systems. In this paper we study the oscillation of the modulus of the Rudin–Shapiro polynomials on the unit circle. We also show that the Rudin–Shapiro polynomials Pk and Qk of degree n−1 with n:=2k have o(n) zeros on the unit circle. This should be compared with a result of Conrey, Granville, Poonen and Soundararajan stating that for odd primes p the Fekete polynomials fp of degree p−1 have asymptotically κ0p zeros on the unit circle, where 0.500813>κ0>0.500668. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by Rodgers. We also prove that there are absolute constants c1>0 and c2>0 such that the kth Rudin–Shapiro polynomials Pk and Qk of degree n−1=2k−1 have at least c2n zeros in the annulus z∈double-struckC:1−c1n<|z|<1+c1n.

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“…Since

f_{p}

has constant coefficient 0, it is not a Littlewood polynomial, but

g_{p}

defined by

g_{p} false(z false) : = f_{p} false(z false) / z

is a Littlewood polynomial of degree

p - 2

. Fekete polynomials are examined in detail in . In the authors examined the maximal size of the Mahler measure and the

L_{p}

norms of sums of n monomials on the unit circle as well as on subarcs of the unit circles.…”

Section: Introduction and Notationmentioning

confidence: 99%

“…Fekete polynomials are examined in detail in [4,22,32,33,37,39,49]. In [18,19] the authors examined the maximal size of the Mahler measure and the L p norms of sums of n monomials on the unit circle as well as on subarcs of the unit circles.…”

mentioning

confidence: 99%