Level sets, a Gauss-Fourier conjecture, and a counter-example to a conjecture of Borcea and Shapiro
The paper consists of two parts. In the first part of the paper the authors construct a counter-example to a conjecture of \textit{J. Borcea} and \textit{B. Shapiro} [Int. Math. Res. Not. 2004, No. 69, 3689--3708 (2004; Zbl 1076.14078)]. To formulate their conjecture we need certain notation. Let \(P\) be a real polynomial of degree \(n\geq 2\). Consider the set \(\mathcal{G}_{P}=\{z\in\mathbb{CP}^{1}:\;P'(z)/P(z)\in\mathbb{R}\}\) and its connected component \(\mathcal{C}\) containing the real axis. A chord is a connected component of \(\mathcal{C}\cap\{z\in\mathbb{C}:\mathrm{Im}(z)>0\}\). The conjecture of Borcea and Shapiro states that each chord contains at least one non-real zero of \(P\). Using level sets the authors give an elegant proof of the fact that for the polynomial \(P(z)=z(z^{2}+1)^{5}\) there exists a chord of \(\mathcal{G}_{P}\) containing no non-real zeros of \(P\). In the second part of the paper the authors prove that for every non-constant real entire function \(f\) the set \(K=\{z:\mathrm{Im}\left(z-(f/f')(z)\right)>0\}\) does not have bounded components. This fact answers a question of \textit{S. Hellerstein} and the first author [Complex Variables, Theory Appl. 47, No. 1, 25--57 (2002; Zbl 1021.30026)]. In addition, the authors present interesting historical information related to the Hawai'i conjecture.
scientific article; zbMATH DE number 5965024