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.
