scholarly journals Regular polygons, Morgan-Voyce polynomials, and Chebyshev polynomials

2021 ◽  
Vol 27 (2) ◽  
pp. 79-87
Author(s):  
Jorma K. Merikoski ◽  
Keyword(s):  
Chebyshev Polynomials ◽  
Monic Polynomial ◽  
Regular Polygons ◽  
Integer Coefficients

We say that a monic polynomial with integer coefficients is a polygomial if its each zero is obtained by squaring the edge or a diagonal of a regular n-gon with unit circumradius. We find connections of certain polygomials with Morgan-Voyce polynomials and further with Chebyshev polynomials of second kind.

Chebyshev Polynomials and Integer Coefficients

2019 ◽  
Vol 105 (1-2) ◽  
pp. 291-300
Author(s):  
R. M. Trigub
Keyword(s):  
Chebyshev Polynomials ◽  
Integer Coefficients

scholarly journals Algebraic Integers as Chromatic and Domination Roots

2012 ◽  
Vol 2012 ◽  
pp. 1-8 ◽  
Author(s):  
Saeid Alikhani ◽  
Roslan Hasni
Keyword(s):  
Monic Polynomial ◽  
Chromatic Polynomial ◽  
Simple Graph ◽  
Dominating Sets ◽  
Algebraic Integers ◽  
Integer Coefficients ◽  
Chromatic Root ◽  
Domination Polynomial

Let G be a simple graph of order n and λ∈ℕ. A mapping f:V(G)→{1,2,…,λ} is called a λ-colouring of G if f(u)≠f(v) whenever the vertices u and v are adjacent in G. The number of distinct λ-colourings of G, denoted by P(G,λ), is called the chromatic polynomial of G. The domination polynomial of G is the polynomial D(G,λ)=∑i=1nd(G,i)λi, where d(G,i) is the number of dominating sets of G of size i. Every root of P(G,λ) and D(G,λ) is called the chromatic root and the domination root of G, respectively. Since chromatic polynomial and domination polynomial are monic polynomial with integer coefficients, its zeros are algebraic integers. This naturally raises the question: which algebraic integers can occur as zeros of chromatic and domination polynomials? In this paper, we state some properties of this kind of algebraic integers.

The Total Character of a Finite Group

2015 ◽  
Vol 22 (spec01) ◽  
pp. 775-788
Author(s):  
Stephen Humphries ◽  
Chelsea Kennedy ◽  
Emma Rode
Keyword(s):  
Finite Group ◽  
Chebyshev Polynomials ◽  
Irreducible Character ◽  
Character Group ◽  
Irreducible Characters ◽  
Integer Coefficients ◽  
A Finite Group

The total character τ(G) of a finite group G is the sum of the irreducible characters of G. We present conditions under which τ(G) can be written as a polynomial with integer coefficients in an irreducible character of G. Such a group we call a total character group. We show that the dicyclic group of order 4n is a total character group if and only if n ≡ 2, 3 mod 4. The polynomial used is a sum of Chebyshev polynomials of the second kind. We also show that Sn (n ≥ 4) is not a total character group.

scholarly journals Total characters and Chebyshev polynomials

2003 ◽  
Vol 2003 (38) ◽  
pp. 2447-2453 ◽  
Author(s):  
Eirini Poimenidou ◽  
Homer Wolfe
Keyword(s):  
Finite Group ◽  
Chebyshev Polynomials ◽  
Irreducible Character ◽  
Dihedral Group ◽  
Dihedral Groups ◽  
Irreducible Characters ◽  
Integer Coefficients ◽  
Complete Answer ◽  
A Finite Group

The total characterτof a finite groupGis defined as the sum of all the irreducible characters ofG. K. W. Johnson asks when it is possible to expressτas a polynomial with integer coefficients in a single irreducible character. In this paper, we give a complete answer to Johnson's question for all finite dihedral groups. In particular, we show that, when such a polynomial exists, it is unique and it is the sum of certain Chebyshev polynomials of the first kind in any faithful irreducible character of the dihedral groupG.

scholarly journals A Commutativity Theorem for Division Rings

1980 ◽  
Vol 23 (2) ◽  
pp. 241-243 ◽  
Author(s):  
Anthony Richoux
Keyword(s):  
Division Ring ◽  
Monic Polynomial ◽  
Commutativity Theorem ◽  
Division Rings ◽  
Integer Coefficients

AbstractLet D be a division ring with center Z. Suppose for all xϵD, there exists a monic polynomial, fx(t), with integer coefficients such that fx(x)ϵZ. Then D is commutative.

scholarly journals Notes on the Distribution of Roots Modulo a Prime of a Polynomial II

2019 ◽  
Vol 14 (1) ◽  
pp. 87-104
Author(s):  
Yoshiyuki Kitaoka
Keyword(s):  
Linear Relation ◽  
Monic Polynomial ◽  
Integer Coefficients ◽  
Congruence Relations ◽  
Distribution Of Roots

AbstractLet f (x) be a monic polynomial with integer coefficients and 0 ≤ r1 ≤ ··· ≤ rn<p its roots modulo a prime p. We generalize a conjecture on the distribution of roots ri with additional congruence relations ri ≡ Ri mod L from the case that f has no non-trivial linear relation among roots to the case that f has a non-trivial linear relation.

Dense Periodic Packings of Regular Polygons

1995 ◽  
Vol 5 (12) ◽  
pp. 1539-1550 ◽  
Author(s):  
Y. Limon Duparcmeur ◽  
A. Gervois ◽  
J. P. Troadec
Keyword(s):  
Regular Polygons

Random Close Packings of Regular Polygons

1997 ◽  
Vol 7 (10) ◽  
pp. 1181-1189 ◽  
Author(s):  
Y. Limon Duparcmeur ◽  
J. P. Troadec ◽  
A. Gervois
Keyword(s):  
Regular Polygons
Sign in / Sign up

Export Citation Format

Share Document