math.CV — Complex Variables
Sheil-Small's (1974) covering problem has a positive answer
T. Sheil-Small asked whether every self-inversive polynomial maps the unit disc onto a domain that contains a disc whose radius is the largest modulus of the polynomial's coefficients (Problem 4.24 in Hayman and Lingham's problem book). We show that the answer is yes. More generally, every non-constant polynomial whose coefficients have the same moduli when read in reverse order maps the unit disc onto a domain that contains an open disc of this radius. This class contains the self-inversive polynomials and the polynomials whose zeros, with multiplicity, are symmetric under taking reciprocals (the problem's printed definition of self-inversive). The radius cannot be increased. For a polynomial with all its zeros on the unit circle, the largest disc in the image has radius exactly half the maximum modulus on the circle. For these polynomials the question is therefore equivalent to the known bound: every coefficient has modulus at most half this maximum (O'Hara-Rodriguez, Saff-Sheil-Small, Kristiansen). The new case is that of polynomials with zeros off the unit circle. We treat this case with the Schur-Cohn Hermitian form and that bound. The main theorem is also proved in the Lean 4 proof assistant, with that bound taken as given.