The nearest polynomial with a zero in a given domain

For a real univariate polynomial f and a closed domain D⊂C whose boundary C is represented by a piecewise rational function, we provide a rigorous method for finding a real univariate polynomial such that has a zero in D and is minimal. First, we prove that if a nearest polynomial exists, there is a nearest polynomial such that the absolute value of every coefficient of is 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.