On the Location of Zeros of an Interval Polynomial

Abstract

For an interval polynomial F, we provide a rigorous method for deciding whether there exists a polynomial in F that has a zero in a prescribed domain D. When D is real, we show that it is sufficient to examine a finite number of polynomials. When D is complex, we assume that the boundary C of D is a simple closed curve of finite length and C is represented by a piecewise rational function. The decision method uses the representation of C and the property that a polynomial in F is of degree one with respect to each coefficient regarded as a variable. Using the method, we can completely determine the set of real numbers that are zeros of a polynomial in F. For complex zeros, we can obtain a set X that contains the set Z(F), which consists of all the complex numbers that are zeros of a polynomial in F, and the difference between X and Z(F) can be as small as possible.