On Littlewood Polynomials with Prescribed Number of Zeros Inside the Unit Disk

2015 ◽  
Vol 67 (3) ◽  
pp. 507-526 ◽  
Author(s):  
Peter Borwein ◽  
Stephen Choi ◽  
Ron Ferguson ◽  
Jonas Jankauskas
Keyword(s):  
Unit Circle ◽  
Prime Number ◽  
Negative Coefficient ◽  
Explicit Formulas ◽  
Limit Values ◽  
Number Of Zeros ◽  
Littlewood Polynomials ◽  
Sign Change ◽  
Complex Zeros ◽  
Littlewood Polynomial

AbstractWe investigate the numbers of complex zeros of Littlewood polynomials p(z) (polynomials with coefficients {−1, 1}) inside or on the unit circle |z| = 1, denoted by N(p) and U(p), respectively. 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. We obtain explicit formulas for N(p), U(p) for polynomials p(z) of these types. We show that if n + 1 is a prime number, then for each integer k, 0 ≤ k ≤ n − 1, there exists a Littlewood polynomial p(z) of degree n with N(p) = k and U(p) = 0. Furthermore, we describe some cases where the ratios N(p)/n and U(p)/n have limits as n → ∞ and find the corresponding limit values.

scholarly journals Unimodular Roots of Special Littlewood Polynomials

2006 ◽  
Vol 49 (3) ◽  
pp. 438-447 ◽  
Author(s):  
Idris David Mercer
Keyword(s):  
Unit Circle ◽  
Minimum Modulus ◽  
Littlewood Polynomials ◽  
Littlewood Polynomial

AbstractWe call α(z) = a0 + a1z + · · · + an–1zn–1 a Littlewood polynomial if aj = ±1 for all j. We call α(z) self-reciprocal if α(z) = zn–1α(1/z), and call α(z) skewsymmetric if n = 2m + 1 and am+j = (–1)jam–j for all j. It has been observed that Littlewood polynomials with particularly high minimum modulus on the unit circle in ℂ tend to be skewsymmetric. In this paper, we prove that a skewsymmetric Littlewood polynomial cannot have any zeros on the unit circle, as well as providing a new proof of the known result that a self-reciprocal Littlewood polynomial must have a zero on the unit circle.

scholarly journals Lq Norms of Fekete and Related Polynomials

2017 ◽  
Vol 69 (4) ◽  
pp. 807-825 ◽  
Author(s):  
Christian Günther ◽  
Kai-Uwe Schmidt
Keyword(s):  
Positive Integer ◽  
Unit Circle ◽  
Finite Fields ◽  
Legendre Symbol ◽  
Littlewood Polynomials ◽  
First Results ◽  
Recursive Formulas ◽  
Littlewood Polynomial ◽  
Limiting Values ◽  
Specific Sequences

AbstractA Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in {−1, 1}. There are various old unsolved problems, mostly due to Littlewood and Erdos, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq normon the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and (· |p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp when q is an even positive integer and p → ∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.

Polynomials with coefficients from a finite set

2014 ◽  
Vol 64 (6) ◽  
Author(s):  
Javad Baradaran ◽  
Mohsen Taghavi
Keyword(s):  
Unit Circle ◽  
Open Disk ◽  
Complex Polynomials ◽  
Number Of Zeros ◽  
Littlewood Polynomials ◽  
Finite Set

AbstractThis paper focuses on the problem concerning the location and the number of zeros of those polynomials when their coefficients are restricted with special conditions. The problem of the number of the zeros of reciprocal Littlewood polynomials on the unit circle $\mathbb{T}$ is discussed, the interest on bounds for the number of the zeros of reciprocal polynomials on the unit circle arose after 1950 when Erdös began introducing problems on zeros of various types of polynomials. Our main result is the problem of finding the number of zeros of complex polynomials in an open disk.

scholarly journals A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path

2012 ◽  
Vol 64 (4) ◽  
pp. 822-844 ◽  
Author(s):  
J. Haglund ◽  
J. Morse ◽  
M. Zabrocki
Keyword(s):  
Building Blocks ◽  
Main Diagonal ◽  
Combinatorial Theory ◽  
Dyck Path ◽  
Dyck Paths ◽  
Littlewood Polynomials ◽  
Diagonal Harmonics ◽  
Littlewood Polynomial ◽  
Theory Operator ◽  
Shuffle Conjecture

Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of q, t-Catalan sequences, and we prove a number of identities involving these functions.

scholarly journals On the Lq norm of cyclotomic Littlewood polynomials on the unit circle

2011 ◽  
Vol 151 (2) ◽  
pp. 373-384 ◽  
Author(s):  
TAMÁS ERDÉLYI
Keyword(s):  
Unit Circle ◽  
Main Tool ◽  
Factorization Theorem ◽  
Cyclotomic Polynomials ◽  
Littlewood Polynomials ◽  
Odd Degree ◽  
Image Position

Let n be the collection of all (Littlewood) polynomials of degree n with coefficients in {−1, 1}. In this paper we prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈ 2ν, then for every q > 2 with some a = a(q) > 1/2 depending only on q, where The case q = 4 of the above result is due to P. Borwein, Choi and Ferguson. We also prove that if (P2ν) is a sequence of cyclotomic polynomials P2ν ∈ 2ν, then for every 0 < q < 2 with some 0 < b = b(q) < 1/2 depending only on q. Similar results are conjectured for Littlewood polynomials of odd degree. Our main tool here is the Borwein–Choi Factorization Theorem.

ALGEBRAIC AND GEOMETRIC PROPERTIES OF LATTICE WALKS WITH STEPS OF EQUAL LENGTH

2016 ◽  
Vol 95 (2) ◽  
pp. 338-346
Author(s):  
KRZYSZTOF KOŁODZIEJCZYK ◽  
RAFAŁ SAŁAPATA
Keyword(s):  
Normal Subgroup ◽  
Prime Number ◽  
Quotient Group ◽  
Prime Numbers ◽  
Lattice Points ◽  
Equal Length ◽  
Explicit Formulas ◽  
Terminal Set ◽  
Lattice Walks ◽  
Algebraic Properties

A lattice walk with all steps having the same length $d$ is called a $d$-walk. Denote by ${\mathcal{T}}_{d}$ the terminal set, that is, the set of all lattice points that can be reached from the origin by means of a $d$-walk. We examine some geometric and algebraic properties of the terminal set. After observing that $({\mathcal{T}}_{d},+)$ is a normal subgroup of the group $(\mathbb{Z}^{N},+)$, we ask questions about the quotient group $\mathbb{Z}^{N}/{\mathcal{T}}_{d}$ and give the number of elements of $\mathbb{Z}^{2}/{\mathcal{T}}_{d}$ in terms of $d$. To establish this result, we use several consequences of Fermat’s theorem about representations of prime numbers of the form $4k+1$ as the sum of two squares. One of the consequences is the fact, observed by Sierpiński, that every natural power of such a prime number has exactly one relatively prime representation. We provide explicit formulas for the relatively prime integers in this representation.

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