Polynomial convexity, special polynomial polyhedra and applications. (Q2499685)

The main result of the paper is the following theorem. Let \(K\Subset\mathbb C^n\) an \(\mathcal L\)-regular polynomially convex compact set. Let \(g_K\) denote the Green function of the complement of \(K\) with pole at infinity. Put \(D(R):=\{z\in\mathbb C^n: g_K(z) < R\}\), \(R > 0\). Then for any \(0<\varepsilon\ll1\) there exist \(d\geq1\) and polynomials \(p_1,\dots,p_n\) of degree \(d\) such that \(\| p_j\| _K\leq1\), \(j=1,\dots,n\), and \(K\subset\overline{D(\varepsilon)}\subset\mathcal P\subset D(\varepsilon+\varepsilon^2)\), where \(\mathcal P\) is the union of all connected components of the set \(\{z\in\mathbb C^n: \frac1d\log| p_j(z)| <\varepsilon+\beta(\varepsilon),\;j=1,\dots,n\}\) that intersects \(\overline{D(\varepsilon)}\), and \(0<\beta(\varepsilon)\leq\varepsilon^2/2\). The mapping \(F=(p_1,\dots,p_n)\) can be chosen so that: (i) the homogeneous component of \(F\) of degree \(d\) has a unique zero at the origin, (ii) \(F\) has a finite set \(Z\) of zeros and each zero is of multiplicity one, (iii) \(\frac{\#(Z\cap\mathcal P)}{d^n}\to1\) when \(\varepsilon\to0\). Moreover, if \(K\) is the closure of a bounded pseudoconvex balanced domain with continuous Minkowski function, then for any \(0<\varepsilon,\delta\ll1\) there exist \((1-\delta)d\leq d'\leq d\) such that the mapping \(F\) can be chosen so that \(\text{ord}_0p_j\geq d'\), \(j=1,\dots,n\).