Local polynomial convexity of graphs of functions in several variables
The author studies the local polynomial convexity of graphs of smooth functions of several complex variables. The main result is the following theorem. Let \(U\) be an open neighborhood of \(0\in\mathbb C^n\) and let \(f\in\mathcal C^2(U,\mathbb C)\). Assume that there exists a \(g\in\mathcal C^2(U,\mathbb C)\) such that: {\parindent=8mm \begin{itemize} \item[(i)] \(|f|^2\leq\text{Re}(fg)\), \item [(ii)] there exists a \(p\in\mathbb N\) such that \(|g(z)|\leq|z|^p\) for \(z\in U\), and the function \((z,w)\longmapsto|w|^2-\text{Re}(wg(z))\) is plurisubharmonic in a neighborhood of \(0\in\mathbb C^{n+1}\), \item [(iii)] \(\Big|\frac{\partial f}{\partial\overline z}(z)\overline w\Big|^2+\text{Re}(f(z)L_{\overline f}(z)(w) )>|g(z)L_{\overline f}(z)(w)|\) for all \(z\in U\setminus\{0\}\) and \(w\in(\mathbb C\setminus\{0\})^n\). \end{itemize}} Then the graph \(\varGamma_f\) of \(f\) is polynomially convex at \(0\in\mathbb C^{n+1}\). Moreover, there exists an \(r>0\) such that each continuous function on \(\varGamma_f\cap B(0,r)\) can be approximated uniformly by polynomials.
scientific article; zbMATH DE number 6485380
scientific article