On a geometric property of Lemniscates
Motivated by a property of polynomials of a complex variable, the authors prove the theorem below and discuses related open questions. Theorem. Let \(p_n(w,w_k) = \prod_{k=1}^n|w-w_k|\quad (w,w_k\in\mathbb R^3)\) and \(E(p_n)=\{w: p_n(w,w_k)\leq1\}\). If \(p_n(w,w_k)\) and \(p_n^{^*}(w,w_k^{^*})\) are such that \(E(p_n)\subseteq E(p_n^{^*})\) and if all the zeros \(w_k\) of \(p_n\) lie on the same plane, then \(p_n(w,w_k)\equiv p_n^{^*}(w,w_k^{^*})\). Moreover, the hypothesis \(E(p_n)\subseteq E(p_n{^*})\) is not sufficient to deduce \(p_n=p_n^{^*}\). [For further properties of products \(p_n(w,w_k)\), see \textit{J.B.Diaz} and \textit{D.B.Schaffer}, Appl. Anal. 6, 109-117 (1977; Zbl 0346.30003).]
scientific article; zbMATH DE number 3595483