On the cyclic resultants
type: Item
instance of: scholarly article
title: On the cyclic resultants
zbMATH Open document ID: 1160.11007
author: J.-P. Bézivin
published in: Proceedings of the Japan Academy. Series A
publication date: 2008-08-11T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.pja/1201012522
review text: In this paper the author gives an alternative elementary proof of a theorem of \textit{C. J. Hillar} [J. Symb. Comput. 39, No. 6, 653--669 (2005); erratum 40, No. 3, 1126--1127 (2005; Zbl 1130.12002)]. Theorem: Let \(K\) be a field of zero characteristic. Let \(P(X)=a_0(X-\lambda_1)\ldots (X-\lambda_d)\) and \(Q(X)=b_0(X-\mu_1)\ldots (X-\mu_{d'})\) be two polynomials in \(K[X]\) such that \(\lambda_i,\mu_i\in K^*\) are not roots of unity. (1) Suppose that \(\mathrm{Res}(P,X^n-1)=\mathrm{Res}(Q,X^n-1)\) for each \(n\geq 1\). Then there exists two polynomials \(U\) and \(V\) with \(U\) is of even degree such that \(P(X)=V(X)U^*(X)\) and \(Q(X)=V(X)U(X)\) where \(U^*\) is the reciprocal polynomial. (2) Suppose that \(\mathrm{Res}(P,X^n-1)=-\mathrm{Res}(Q,X^n-1)\) for each \(n\geq 1\). Then there exists two polynomials \(U\) and \(V\) with \(U\) is of odd degree such that \(P(X)=-V(X)U^*(X)\) and \(Q(X)=V(X)U(X)\) where \(U^*\) is the reciprocal polynomial.
reviewed by: Ali Benhissi
Mathematics Subject Classification ID: 11B37
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 5309655
zbMATH Keywords: Cyclic resultant
zbMATH Keywords: Recurring sequences
zbMATH Keywords: Polynomials
MaRDI profile type: Publication
OpenAlex ID: W1970183378
cites work: Q3817302
cites work: Cyclic resultants
cites work: Polynomial recurrences and cyclic resultants
cites work: Quantum computation of zeta functions of curves
DOI: 10.3792/PJAA.83.157
description: scientific article; zbMATH DE number 5309655