A counterexample to the complex polynomial Koebe radius conjecture

Abstract

We disprove the complex-coefficient polynomial Koebe radius conjecture proposed by Dmitrishin, Smorodin and Stokolos in 2018. We give a normalised cubic with rational real and imaginary coefficients that is univalent on a neighbourhood of the closed unit disk and omits a point of modulus smaller than (3−√5)/2. Together with the known real cubic theorem, this separates the real and complex cubic Koebe radii. Degree three is the least possible degree of failure. The proof specialises the classical cubic univalence criterion and certifies its strict inequalities by exact rational Bernstein coefficients.