The nearest polynomial with a zero in a given domain from a geometrical viewpoint

Abstract

For a real univariate polynomial f and a closed complex domain D, whose boundary C is a simple curve parameterized by a univariate piecewise rational function, a rigorous method is given for finding a real univariate polynomial f such that f has a zero in D and |f - ~f|∞ is minimal. First, it is proved that the minimum distance between f and polynomials having a zero at α in C is a piecewise rational function of the real and imaginary parts of α. Thus, on C, the minimum distance is a piecewise rational function of a parameter obtained through the parameterization of C. Therefore, by using the property that ~f has a zero on C and computing the minimum distance on C, ~f can be constructed.