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Zeros of certain cyclotomy-generated polynomials. (English) Zbl 0726.11015
The “cyclotomy-generated polynomial” is defined for \(n\geq 1\) by \[ p_ n(x)=x^{n^ 2+n}-\Phi_{n^ 2+n}-x^{2n+1}\Phi_{n^ 2- 1}/\Phi_{n+1}+x^{2n}\Phi_{n^ 2-n}/\Phi_ n, \] where \(\Phi_ n(x)=x^{n-1}+x^{n-2}+...+x+1\) and \(\Phi_ 0(x)=0.\)
The zeros of the polynomials \(p_ n(x)\) were studied by A. F. Horadam and A. G. Shannon [Fibonacci numbers and their applications, Pap. 1st Int. Conf., Patras/Greece 1984; Math. Appl. 28, 81-97 (1986; Zbl 0593.12007)] and some conjectures were stated in that article. The author mentions some statement about zeros of \(p_ n(x)\) and solves conjectures from that paper. It is shown e.g.:
1) The numbers \(z_ k=\exp (2\pi ik/n(n+1))\) are zeros of \(p_ n(z)\) for \(1\leq k\leq n^ 2+n-1\), k differs from multiples of n and \(n+1.\)
2) The remaining zeros of \(p_ n(z)\) (including double zero at \(z=1)\) are those of \(f_ n(z)=z^{2n+2}-3z^{2n+1}+z^{2n}+z^{n+1}+z^ n- 1=(z-1)^ 2r_ n(z)\), and the zeros of \(r_ n(z)\) lie outside the circle \(| z| =(1/3)^{1/n}.\)
3) 2n-1 zeros of \(r_ n(z)\) lie within the unit circle.
4) \(r_ n(z)\) has at least one zero in each sector \(| \arg z- (k/n)\pi | \leq \pi /(n+1)\), \(k=0,1,...,2n-1\).
Reviewer: L.Skula (Brno)

11B83 Special sequences and polynomials
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
11C08 Polynomials in number theory