Vol. 89, No. 1, 1980

Recent Issues
Vol. 332: 1
Vol. 331: 1  2
Vol. 330: 1  2
Vol. 329: 1  2
Vol. 328: 1  2
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Online Archive
Volume:
Issue:
     
The Journal
About the journal
Ethics and policies
Peer-review process
 
Submission guidelines
Submission form
Editorial board
Officers
 
Subscriptions
 
ISSN 1945-5844 (electronic)
ISSN 0030-8730 (print)
 
Special Issues
Author index
To appear
 
Other MSP journals
On the zeros of convex combinations of polynomials

Harriet Jane Fell

Vol. 89 (1980), No. 1, 43–50
Abstract

Given monic n-th degree polynomials P0(z) and P1(z), let PA(z) = (1 A)P0(z) + AP1(z). If the zeros of P0 and P1 all lie in a circle 𝒞 or on a line L, necessary and sufficient conditions are given for the zeros of PA (0 A 1) to all lie on 𝒞 or L. This describes certain convex sets of monic n-th degree polynomials having zeros in 𝒞 or L. If the zeros of P0 and P1 lie in the unit disk and P0 and P1 have real coefficients, then the zeros of PA (0 A 1) lie in the disk |z| < cos(π∕2n)sin(π∕2n). A set is described which includes the locus of zeros of PA (0 A 1) as P0 and P1 vary through all monic n-th degree polynomials having all their zeros in a compact set K. When K is path-connected, this locus is exactly the set described.

Mathematical Subject Classification 2000
Primary: 30C15
Milestones
Received: 21 March 1979
Revised: 28 August 1979
Published: 1 July 1980
Authors
Harriet Jane Fell