Volume of polynomial lemniscates in \(\mathbb C^n\)
Let \(P\) be a polynomial of \(n\) complex variables of degree \(m\). Define \(E_r:=\{z=(z_1,\dots,z_n)\in\mathbb C^n: | z_j| \leq r\), \(j=1,\dots,n\}\), \(E(P;r,\varepsilon):=\{z\in E_r: | P(z)| \leq\varepsilon^m\}\), \(r,\varepsilon>0\). Assume that \(\max_{E_r}| P| =1\). The author studies upper and lower bounds for \(\lambda(E(P;r,\varepsilon))\), where \(\lambda\) is the \(2n\)--dimensional Lebesgue measure. He shows that for every \(\alpha<2\) there exists a \(C_\alpha>0\) such that \(\lambda(E(P;r,\varepsilon))\leq C_\alpha r^{2n}\varepsilon^\alpha\). On the other hand, if \(\varepsilon<1/2\sqrt{n}\), then \(\lambda(E(P;r,\varepsilon))\geq\lambda(B)2^{-2n}r^{2n}\varepsilon^{2nm/k(r)}\), where \(B\) is the unit Euclidean ball in \(\mathbb C^n\), \(k(r):=\max\{k_P(z): z\in E_r\}\), and \(k_P(z)\) is the order of zero of \(P\) at \(z\).
polynomial lemniscates in \(\mathbb C^n\)
scientific article; zbMATH DE number 1992798