Journal of Inequalities and Applications
Volume 4 (1999), Issue 1, Pages 83-89

On the paths to the zeros of a polynomial

Bernard Beauzamy

Société de Calcul Mathématique, S. A. 111, Faubourg Saint Honoré, Paris 75008, France

Received 5 September 1998; Revised 7 October 1998

Copyright © 1999 Bernard Beauzamy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let P(z) be a polynomial in one complex variable, with complex coefficients, and let z1,,zn be its zeros. Assume, by normalization, that P(0)=1. The direct path from 0 to the root zj is the set {P(tzj),0t1}. We are interested in the altitude of this path, which is |P(tzj)| . We show that there is always a zero towards which the direct path declines near 0, which means |P(tzj)|<|P(0)| if t is small enough. However, starting with degree 5, there are polynomials for which no direct path constantly remains below the altitude 1.