These are the extended notes of the plenary lecture on the conference Constructive Functions 2014, Nashville, TN, USA. It deals with the problem how much zeros on the boundary of a set raise the norm of polynomials compared to the minimal norms.
Let \(C_1\) be the unit circle, and recall that if \(P_n(z)=a_n z^n+\cdots+1\) is a polynomial, then \(\|P_n\|_{C_1}\geq 1\), where \(\|f\|_E=\sup_{z\in E}|f(z)|\). Moreover, if \(P_n\) has a zero somewhere on the unit circle, it can be shown that \[ \|P_n\|_{C_1}\geq 1+\frac{1}{8\pi n}\, . \] In the opposite direction G. Halász [Studies in Pure Mathematics, 259–269 (1983; Zbl 0529.10041)] proved that there is a \(P_n(z)\) with \(P_n(1)=0\) such that \[ \|P_n\|_{C_1}\leq e^{2/n}\, , \] and he asked to determine \[ \mu_n=\inf_{P_n(0)=1,\, P_n(1)=0}\|P_n\|_{C_1}\, . \] This problem was solved in the paper [Math. Z. 168, 105–116 (1979; Zbl 0393.30004)] by M. Lachance et al., who proved the formula \[ \mu_n=\cos\left(\frac{\pi}{2(n+1)}\right)^{-n-1}\, . \] The topics in this paper are related to the previous formula. Section 2 briefly describes what happens if there are more than one zero on the unit circle. Sections 3 and 4 explain how the results formulated for the unit circle in Section 2 have extensions to Jordan curves. Section 5 is devoted to some classical discrepancy theorems, which are related to the problem discussed in the paper. Section 6 shows a problem of Erdős while Section 7 deals with high-order zeros and incomplete polynomials.
For the entire collection see [Zbl 1343.00038].