The nearest polynomial with a zero in a given domain

For a real univariate polynomial f and a bounded closed domain D ⊂ C whose boundary C is a simple closed curve of finite length and is represented by a piecewise rational function, we provide a rigorous method for finding the real univariate polynomial f such that f has a zero in D and ||f -- f||∞ is minimal. First, we prove that the absolute value of every coefficient of f -- f is ||f -- f∞ with at most one exception. Using this property and the representation of C, we reduce the problem to solving systems of algebraic equations, each of which consists of two equations with two variables. Furthermore, every equation is of degree one with respect to one of the two variables.