On the zeros of Gonchar polynomials (Q2839360)

For \(d \in \mathbb N\), the Gonchar polynomial \(G_d\) is defined by NEWLINE\[NEWLINE G_d(z) := [(z-1)^d-z-1]z^{d-1}+(z-1)^d\,. NEWLINE\]NEWLINE NEWLINEThe zeros of these polynomials have been investigated to a certain extent by \textit{J. S.~Brauchart, P. D.~Dragnev, E. B.~Saff} and \textit{C. E.~van~de~Woestijne} [Acta. Math. Hungar. 137, No. 1--2, 10--26 (2012)]. For example, they have shown that the largest positive zero \(\xi(d)\) of \(G_d\) lies in the interval \((2,3]\), and they conjectured that the sequence \((\xi(d))\) is strictly decreasing. The first result in this paper states that this is really true. Furthermore, Brauchart et al. conjectured that the zeros of \(G_d\) all lie in one of the three sets \(A_1 := \left\{\, z \in \mathbb C : \text{Re}\,{z}<\frac{1}{2}, \;|z-1|>1 \,\right\}\), \(A_2 := \left\{\, z \in \mathbb C : \text{Re}\,{z}<\frac{1}{2}, \;|z-1|<1 \,\right\}\) and \(A_3 := \left\{\, z \in \mathbb C : \text{Re}\,{z}>\frac{1}{2}, \;|z-1|>1 \,\right\}\) and the numbers \(N_1(d)\), \(N_2(d)\), \(N_3(d)\) of zeros of \(G_d\) in these sets, respectively, follow a very regular pattern. The author proves that this is also true.