On the cyclic resultants (Q935597)

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.