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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)

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