An Application of Newton's Identities: 11096

2006 ◽  
Vol 113 (5) ◽  
pp. 463
Author(s):  
Said Amghibech ◽  
José Heber Nieto
Keyword(s):  
Newton’S Identities

scholarly journals A NOTE ON BARKER POLYNOMIALS

2013 ◽  
Vol 09 (03) ◽  
pp. 759-767 ◽  
Author(s):  
PETER BORWEIN ◽  
TAMÁS ERDÉLYI
Keyword(s):  
Newton’S Identities ◽  
Polynomial Formula ◽  
Barker Codes ◽  
Elegant Proof

We call the polynomial [Formula: see text] a Barker polynomial of degree n-1 if each aj ∈{-1, 1} and [Formula: see text] Properties of Barker polynomials were studied by Turyn and Storer thoroughly in the early sixties, and by Saffari in the late eighties. In the last few years P. Borwein and his collaborators revived interest in the study of Barker polynomials (Barker codes, Barker sequences). In this paper we give a new proof of the fact that there is no Barker polynomial of even degree greater than 12, and hence Barker sequences of odd length greater than 13 do not exist. This is intimately tied to irreducibility questions and proved as a consequence of the following new result. Theorem.Ifn ≔ 2m + 1 > 13and[Formula: see text]where eachbj ∈{-1, 0, 1}for even values of j, each bj is an integer divisible by 4 for odd values of j, then there is no polynomial[Formula: see text]such that[Formula: see text], where[Formula: see text]and[Formula: see text]denotes the collection of all polynomials of degree 2m with each of their coefficients in {-1, 1}. A clever usage of Newton's identities plays a central role in our elegant proof.

scholarly journals The New New-Nacci Method for Calculating the Roots of a Univariate Polynomial and Solution of Quintic Equation in Radicals

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 746
Author(s):  
Ilija Tanackov ◽  
Ivan Pavkov ◽  
Željko Stević
Keyword(s):  
Impossibility Theorem ◽  
Second Phase ◽  
Polynomial Roots ◽  
Real Roots ◽  
Quintic Equation ◽  
Complex Roots ◽  
Abel Polynomial ◽  
Quintic Polynomial ◽  
Univariate Polynomial ◽  
Newton’S Identities

An arbitrary univariate polynomial of nth degree has n sequences. The sequences are systematized into classes. All the values of the first class sequence are obtained by Newton’s polynomial of nth degree. Furthermore, the values of all sequences for each class are calculated by Newton’s identities. In other words, the sequences are formed without calculation of polynomial roots. The New-nacci method is used for the calculation of the roots of an nth-degree univariate polynomial using radicals and limits of successive members of sequences. In such an approach as is presented in this paper, limit play a catalytic–theoretical role. Moreover, only four basic algebraic operations are sufficient to calculate real roots. Radicals are necessary for calculating conjugated complex roots. The partial limitations of the New-nacci method may appear from the decadal polynomial. In the case that an arbitrary univariate polynomial of nth degree (n ≥ 10) has five or more conjugated complex roots, the roots of the polynomial cannot be calculated due to Abel’s impossibility theorem. The second phase of the New-nacci method solves this problem as well. This paper is focused on solving the roots of the quintic equation. The method is verified by applying it to the quintic polynomial with all real roots and the Degen–Abel polynomial, dating from 1821.

Newton's Identities Once Again!

2003 ◽  
Vol 110 (3) ◽  
pp. 232 ◽  
Author(s):  
Jan Minac
Keyword(s):  
Newton’S Identities
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