Unimodular Roots of Reciprocal Littlewood Polynomials

Drungilas, Paulius

doi:10.4134/jkms.2008.45.3.835

Cited by 16 publications

(13 citation statements)

References 7 publications

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“…This was later strengthened by Mukunda [29], who showed that odd degree polynomials of this type have at least 3 roots. Drungilas [11] Our results show that potential counter-example to the conjecture for coefficients in {−1, +1} must have a very particular periodic form, as in Theorem 3…”

Section: A Structure Theorem For Trigonometric Polynomials With Few Rmentioning

confidence: 65%

Counting zeros of cosine polynomials: On a problem of Littlewood

Sahasrabudhe

2019

Advances in Mathematics

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We show that if A is a finite set of non-negative integers then the number of zeros of the function f A (θ) = a∈A cos(aθ), in [0, 2π], is at least (log log log |A|) 1/2−ε . This gives the first unconditional lower bound on a problem of Littlewood and solves a conjecture of Borwein, Erdélyi, Ferguson and Lockhart. We also prove a result that applies to more general cosine polynomials with few distinct rational coefficients. One of the main ingredients in the proof is perhaps of independent interest: we show that if f is an exponential polynomial with few distinct integer coefficients and f "correlates" with a low-degree exponential polynomial P , then f has a very particular structure. This result allows us to prove a structure theorem for trigonometric polynomials with few zeros in [0, 2π] and restricted coefficents.is well known that any continuous, even function f : R/(2πZ) → R can be uniformly approximated by cosine polynomials. Thus, for large N , the behavior of such polynomials is essentially as rich as the class of continuous functions. In what follows, we consider what can be said about the behavior of the function f if the coefficients are restricted to a small set.One of the early champions of questions about polynomials with restricted coefficients was J.E.Littlewood, who studied many problems of this general type and stimulated the field with a wealth of problems [23,24,25,26]. Perhaps most famously, Littlewood asked for the minimum L 1 -norm attained by a function of the form a∈A cos(aθ), where A ⊆ N ∪ {0} is of a given size. This question was resolved (up to constants) in 1981 independently by Konyagin [22], and McGehee, Pigno, and Smith [28] after a series of partial results were obtained by Salem, Cohen, Davenport and Pichorides [33,8,9,32]. We note that these estimates have seen numerous applications and modifications in recent years.Counting zeros of polynomials with restricted coefficients has also received considerable attention;after the seminal work of Littlewood and Offord [27], several results have been proved on the number of roots of "typical" polynomials with restricted coefficients. For example, Kac [20,21] has given an exact integral formula for the expected number of roots of a degree N polynomial with coefficients sampled identically and independently from a Gaussian distribution. In a similar vein, Erdős and Offord [15] showed that almost all polynomials of the form N i=0 ε i x i , where ε 0 , . . . , ε N ∈ {−1, +1}, have ( 2 π + o(1)) log N real roots. These results have been subsequently refined and extended by various authors. For more, we refer the reader to the recent paper of Hoi Nguyen, Oanh Nguyen and Van Vu [30].The study of extremal questions on the number of real zeros of polynomials with restricted coefficients began with the work of Bloch and Pólya [2] in 1932, who studied the maximum number of real zeros attainable by polynomials of the form N i=0 ε i x i , ε 0 , . . . , ε N ∈ {−1, 0, +1}. After improvements by other authors [34,35], Erdős and Turán proved a landmark ...

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Section: A Structure Theorem For Trigonometric Polynomials With Few Rmentioning

confidence: 65%

Counting zeros of cosine polynomials: On a problem of Littlewood

Sahasrabudhe

2019

Advances in Mathematics

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“…studied the distribution of the zeros of PSR polynomials with a small perturbation in their coefficients. Real SR polynomials of height 1 -namely, special cases of Littlewood, Newman and Borwein polynomials -were studied by several authors, see [27][28][29][30][31][32][33][34][35] and references therein 2 . Zeros of the so-called Ramanujan Polynomials and generalizations were analyzed in [37][38][39].…”

Section: Real Self-reciprocal Polynomialsmentioning

confidence: 99%

Polynomials with Symmetric Zeros

Vieira

2019

Polynomials - Theory and Application

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Polynomials whose zeros are symmetric either to the real line or to the unit circle are very important in mathematics and physics. We can classify them into three main classes: the self-conjugate polynomials, whose zeros are symmetric to the real line; the self-inversive polynomials, whose zeros are symmetric to the unit circle; and the self-reciprocal polynomials, whose zeros are symmetric by an inversion with respect to the unit circle followed by a reflection in the real line. Real selfreciprocal polynomials are simultaneously self-conjugate and self-inversive so that their zeros are symmetric to both the real line and the unit circle. In this survey, we present a short review of these polynomials, focusing on the distribution of their zeros.

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“…Mukunda [58] improved this result by showing that every self-reciprocal Littlewood polynomial of odd degree has at least 3 zeros on the unit circle. Drungilas [21] proved that every self-reciprocal Littlewood polynomial of odd degree n ≥ 7 has at least 5 zeros on the unit circle and every self-reciprocal Littlewood polynomial of even degree n ≥ 14 has at least 4 zeros on the unit circle. In [4] two types of Littlewood polynomials are considered: Littlewood polynomials with one sign change in the sequence of coefficients and Littlewood polynomials with one negative coefficient, and the numbers of the zeros such Littlewood polynomials have on the unit circle and inside the unit disk, respectively, are investigated.…”

Section: Introduction and Notationmentioning

confidence: 99%

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

Erdélyi

2019

Acta Arith.

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Let n 1 < n 2 < · · · < n N be non-negative integers. In a private communication Brian Conrey asked how fast the number of real zeros of the trigonometric polynomials T N (θ) = N j=1 cos(n j θ) tends to ∞ as a function of N . Conrey's question in general does not appear to be easy. Let P n (S) be the set of all algebraic polynomials of degree at most n with each of their coefficients in S. For a finite set S ⊂ C let M = M (S) := max{|z| : z ∈ S}. It has been shown recently that if S ⊂ R is a finite set and (P n ) is a sequence of self-reciprocal polynomials P n ∈ P n (S) with |P n (1)| tending to ∞, then the number of zeros of P n on the unit circle also tends to ∞. In this paper we show that if S ⊂ Z is a finite set, then every self-reciprocal polynomial P ∈ P n (S) has at least c(log log log |P (1)|) 1−ε − 1 zeros on the unit circle of C with a constant c > 0 depending only on ε > 0 and M = M (S). Our new result improves the exponent 1/2 − ε in a recent result by Julian Sahasrabudhe to 1 − ε. Sahasrabudhe's new idea [66] is combined with the approach used in [34] offering an essencially simplified way to achieve our improvement. We note that in both Sahasrabudhe's paper and our paper the assumption that the finite set S contains only integers is deeply exploited.

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Unimodular Roots of Reciprocal Littlewood Polynomials

Cited by 16 publications

References 7 publications

Counting zeros of cosine polynomials: On a problem of Littlewood

Counting zeros of cosine polynomials: On a problem of Littlewood

Polynomials with Symmetric Zeros

Improved lower bound for the number of unimodular zeros of self-reciprocal polynomials with coefficients in a finite set

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