Un exemple de disque polynômialement convexe. (An example of a polynomially convex disc)
Let f(z) be a complex valued \(C^ 1\)-function defined in a neighborhood of the closed unit disc \(D\subset {\mathbb{C}}\) and let \(\Delta:=\{(z,f(z))\), \(z\in D\}\). Then \(\Delta\) is said to be polynomially convex if there exists, for \(\forall y\in {\mathbb{C}}^ 2\setminus \Delta\) a polynomial P on \({\mathbb{C}}^ 2\) such that \(| P(y)| >_{x\in \Delta}| P(x)|.\) In this paper, the author proves affirmatively a conjecture by \textit{K. J. Preskenis} [Trans. Am. Math. Soc. 171, 445-467 (1972; Zbl 0257.30037)], namely he obtains: Theorem: Assume that locally f is orientation reversing (i.e. \(| f_{\bar z}(a)| >| f_ z(a)|)\) for \(\forall a\in D\), then \(\Delta\) is polynomially convex.
scientific article; zbMATH DE number 4071333