Zentralblatt MATH
Publications of (and about) Paul Erdös
Zbl.No: 383.30001
Autor: Erdös, Paul; Hwang, J.S.
Title: On a geometric property of Lemniscates. (In English)
Source: Aequationes Math. 17, 344-347 (1978).
Review: Motivated by a property of polynomials of a complex variable, the authors prove the theorem below and discuses related open questions. Theorem. Let pn(w,wk) = prodk = 1n|w-wk| (w,wk in R3) and E(pn) = {w: pn(w,wk) \leq 1}. If pn(w,wk) and pn^*(w,wk^*) are such that E(pn)\subseteq E(pn^*) and if all the zeros wk of pn lie on the same plane, then pn(w,wk)\equiv pn^*(w,wk^*). Moreover, the hypothesis E(pn)\subseteq E(pn{^*}) is not sufficient to deduce pn = pn^*. [For further properties of products pn(w,wk), see J.B.Diaz and D.B.Schaffer, Appl. Anal. 6, 109-117 (1977; Zbl 346.30003).]
Reviewer: A.Giroux
Classif.: * 30C10 Polynomials (one complex variable)
© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag