Vol. 89, No. 1, 1980

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On the zeros of convex combinations of polynomials

Harriet Jane Fell

Vol. 89 (1980), No. 1, 43–50

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
Received: 21 March 1979
Revised: 28 August 1979
Published: 1 July 1980
Harriet Jane Fell