Some properties of extremal polynomials for the Ilieff conjecture

Let Pn denote the family of complex polynomials each of degree n, with leading coefficient 1, and having all of its roots in J9(0,1), the closed unit disc with center at 0 and radius 1. Let p€Pn have roots zly •••, zn and have roots wu •••, wn-ι. For such p we use I(zj), I(p), and I(Pn) to denote the numbers min {|z$—wk\: l^k^n—l},m3x{I(Zj):l^j^n}, and sup {/(/>): psPn} respectively. Then p€Pn is called an extremal polynomial for the Ilieff conjecture if I(p)=I(Pn). With this notation the Gauss-Lucas theorem implies that I(Pn)^2 and the conjecture of Ilieff is that I(p)^l for all psPnWe show that there exist extremal polynomials, that an extremal 'polynomial must have at least one root on each subarc of the unit circle of length i^π, and we find the extremal polynomials for n—Z and 4. We begin with a