Let \({\mathcal G}_n\) be the set of polynomials \(P(z)= \sum^n_0 a_j z^j\), where the \(a_j\) are complex numbers of absolute value 1. The authors announce the following striking results: Theorem 1. There are \(P(z)\in {\mathcal G}_n\) depending on the parameter \(q\), such that \(|P_n'|_q\approx n|P_n|_q\) as \(n\) tends to infinity. Here, \(q\) can take every positive value with the exception of 2. By Bernstein’s inequality the number \(n\) can not be replaced by an essentially larger factor.
Theorem 2. If \(P\in {\mathcal G}_n\) and \(2< q< \infty\) and \(|P|_q/(n+ 1)^{1/2}< C\), then for every \(s\) in \((1, q]\) there is a constant \(\theta= \theta(q, s, C)< 1\) such that \(|P'|_s\leq \theta n|P|_s\). The theorem is false for \(q= \infty\), it remains an open question whether it may also be true for \(0< q\leq 1\). J. E. Littlewood raised the question whether there are ‘ultraflat’ polynomials \(P\in {\mathcal G}_n\) for which \[ (1- \varepsilon_n) \sqrt{n+ 1}< |P(e^{it})|< (1+ \varepsilon_n) \sqrt{n+ 1} \] for all real \(t\) and for \(\varepsilon_n\to 0\). The authors announced the astonishingly precise
Theorem 3. Let \(\gamma_q= (1+ q)^{- 1/q}\) \((0< q< \infty)\), \(\gamma_\infty= 1\); \(\varepsilon_n= Kn^{- 1/n^{96}}\). If \(0< \eta\leq \infty\) one can find \(P\in {\mathcal G}_n\) such that \[ |P'|_q= \gamma_q n^{n^{3/2}}+ O_\eta(n^{3/2} \varepsilon_n)\qquad (n\geq 1). \] The authors state an equivalent version of Theorem 3 expressed in terms of \(\omega(t)= \arg P(e^{it})\) from which it follows that the distribution of \(\omega'(t)/n\) approaches a uniform distribution as \(n\to \infty\) in any subinterval of \((- \pi, \pi)\).