scholarly journals On Newman polynomials which divide no Littlewood polynomial

2009 ◽  
Vol 78 (265) ◽  
pp. 327-327 ◽  
Author(s):  
Artūras Dubickas ◽  
Jonas Jankauskas
Keyword(s):  
Newman Polynomials ◽  
Littlewood Polynomial

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.

Large Newman polynomials

1986 ◽  
pp. 159-170
Author(s):  
David W. Boyd
Keyword(s):  
Newman Polynomials

scholarly journals Negative Pisot and Salem numbers as roots of Newman polynomials

2014 ◽  
Vol 44 (1) ◽  
pp. 113-138 ◽  
Author(s):  
Kevin G. Hare ◽  
Michael J. Mossinghoff
Keyword(s):  
Salem Numbers ◽  
Newman Polynomials

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.

CYCLOTOMIC FACTORS OF BORWEIN POLYNOMIALS

2019 ◽  
Vol 100 (1) ◽  
pp. 41-47
Author(s):  
BISWAJIT KOLEY ◽  
SATYANARAYANA REDDY ARIKATLA
Keyword(s):  
Unit Circle ◽  
Prime Divisor ◽  
Monic Polynomial ◽  
Cyclotomic Polynomial ◽  
Newman Polynomials

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

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 Combinatorial Formula for the Hilbert Series of bigraded $S_n$-modules

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Meesue Yoo
Keyword(s):  
Hilbert Series ◽  
Macdonald Polynomials ◽  
Weighted Sum ◽  
Young Tableaux ◽  
Combinatorial Formula ◽  
Nous Proposons ◽  
International Audience ◽  
Standard Young Tableaux ◽  
Littlewood Polynomial ◽  
The Way

International audience We introduce a combinatorial way of calculating the Hilbert series of bigraded $S_n$-modules as a weighted sum over standard Young tableaux in the hook shape case. This method is based on Macdonald formula for Hall-Littlewood polynomial and extends the result of $A$. Garsia and $C$. Procesi for the Hilbert series when $q=0$. Moreover, we give the way of associating the fillings giving the monomial terms of Macdonald polynomials to the standard Young tableaux. Nous introduisons une méthode combinatoire pour calculer la série de Hilbert de modules bigradués de $S_n$ comme une somme pondérée sur les tableaux de Young standards à la forme crochet. Cette méthode se fonde sur la formule Macdonald pour les polynômes Hall-Littlewood et généralise un résultat de $A$. Garsia et $C$. Procesi pour la série de Hilbert dans le cas $q=0$. De plus, nous proposons une méthode pour associer aux tableaux de Young standards les remplissages des monômes des polynômes de Macdonald.

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.

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