Radii of stability and instability for polynomials of degree three and four (Q1204606)

The set of polynomials \(P_ n\) of the form \(a(x)=a_ 0+ a_ 1 x+\cdots + a_ n x^ n\) are considered. Formulas are derived for the stability and instability radii \(R_ S(a)=\inf_{b\in U}\| a-b\|\) and \(R_ U(a)=\inf_{b\in S}\| a-b\|\), respectively (\(S\) and \(U\) being the sets of stable and instable polynomials, respectively), for the case of the polynomials \(P_ 3\) and \(P_ 4\). The formulas are proved.