Multiple Roots of [−1, 1] Power Series

Beaucoup, Franck; Borwein, Peter; Boyd, David W.; Pinner, Christopher

doi:10.1112/s0024610798005857

Cited by 44 publications

(54 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The points of V L inside the unit circle are related to the vanishing points of power series with ±1 coefficients. Beaucoup, Borwein, Boyd and Pinner studied the extremal zeros of such power series and their multiplicity in [2] and [3].…”

Section: Introductionmentioning

confidence: 99%

On Newman polynomials which divide no Littlewood polynomial

Dubickas

Jankauskas

2009

Math. Comp.

View full text Add to dashboard Cite

Abstract. Recall that a polynomial P (x) ∈ Z[x] with coefficients 0, 1 and constant term 1 is called a Newman polynomial, whereas a polynomial with coefficients −1, 1 is called a Littlewood polynomial. Is there an algebraic number α which is a root of some Newman polynomial but is not a root of any Littlewood polynomial? In other words (but not equivalently), is there a Newman polynomial which divides no Littlewood polynomial? In this paper, for each Newman polynomial P of degree at most 8, we find a Littlewood polynomial divisible by P . Moreover, it is shown that every trinomial 1 + ux a + vx b , where a < b are positive integers and u, v ∈ {−1, 1}, so, in particular, every Newman trinomial 1 + x a + x b , divides some Littlewood polynomial. Nevertheless, we prove that there exist Newman polynomials which divide no Littlewood polynomial, e.g., x 9 +x 6 +x 2 +x+1. This example settles the problem 006:07 posed by the first named author at the 2006 West Coast Number Theory conference. It also shows that the sets of roots of Newman polynomials V N , Littlewood polynomials V L and {−1, 0, 1} polynomials V are distinct in the sense that between them there are only trivial relationsThe proofs of several main results (after some preparation) are computational.

show abstract

Section: Introductionmentioning

confidence: 99%

On Newman polynomials which divide no Littlewood polynomial

Dubickas

Jankauskas

2009

Math. Comp.

View full text Add to dashboard Cite

show abstract

“…We mentioned in the beginning of this paper that there exist various results of multiple roots [3,4]. We say that a z ∈ C is a multiple root of a holomorphic function f of order k if for all integers i = 0, 1, 2, .…”

Section: Some Further Problemsmentioning

confidence: 99%

“…Since then many related works have appeared, most notable amongst these are the number theoretic results of Beaucoup, Borwein, Boyd and Pinner [3], and Borwein, Erdélyi and Littmann [4], who studied the distribution of roots and multiple roots. Related work also appeared in Bousch [5], where it was shown that R({−1, 1}) is dense in {z : |z| 4 ∈ [1/2, 2]}.…”

Section: Introductionmentioning

confidence: 99%

Root Sets of Polynomials and Power Series with Finite Choices of Coefficients

Baker

2017

Comput. Methods Funct. Theory

View full text Add to dashboard Cite

Given H ⊆ C two natural objects to study are the set of zeros of polynomials with coefficients in H ,and the set of zeros of a power series with coefficients in H ,In this paper, we consider the case where each element of H has modulus 1. The main result of this paper states that for any r ∈ (1/2, 1), if H is 2 cos −1 ()-dense in S 1 , then the set of zeros of polynomials with coefficients in H is dense in {z ∈ C : |z| ∈ [r, r −1 ]}, and the set of zeros of power series with coefficients in H contains the annulus {z ∈ C : |z| ∈ [r, 1)}. These two statements demonstrate

show abstract

“…The set M has connections to Bernoulli convolutions, Dirichlet forms on fractals, and zeros of polynomials with integer coefficients. See for example [6,9,39,45,[110][111][112]114]. To see the relationship between zeros of polynomials with integer coefficients and M, note that finite compositions f σ |k (z) of the maps in F λ give rise to polynomials of the form…”

Section: Mandelbrot Set For Pairs Of Linear Mapsmentioning

confidence: 99%

Developments in fractal geometry

Barnsley¹,

Vince²

2013

Bull. Math. Sci.

View full text Add to dashboard Cite

Iterated function systems have been at the heart of fractal geometry almost from its origins. The purpose of this expository article is to discuss new research trends that are at the core of the theory of iterated function systems (IFSs). The focus is on geometrically simple systems with finitely many maps, such as affine, projective and Möbius IFSs. There is an emphasis on topological and dynamical systems aspects. Particular topics include the role of contractive functions on the existence of an attractor (of an IFS), chaos game orbits for approximating an attractor, a phase transition to an attractor depending on the joint spectral radius, the classification of attractors according to fibres and according to overlap, the kneading invariant of an attractor, the Mandelbrot set of a family of IFSs, fractal transformations between pairs of attractors, tilings by copies of an attractor, a generalization of analytic continuation to fractal functions, and attractor-repeller pairs and the Conley "landscape picture" for an IFS.

show abstract

Multiple Roots of [−1, 1] Power Series

Cited by 44 publications

References 0 publications

On Newman polynomials which divide no Littlewood polynomial

On Newman polynomials which divide no Littlewood polynomial

Root Sets of Polynomials and Power Series with Finite Choices of Coefficients

Developments in fractal geometry

Contact Info

Product

Resources

About