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)

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