Some properties of zeros of polynomials with vanishing coefficients (Q1014462)

Various questions concerning zeros of complex polynomials whose coefficients are continuous functions of a real parameter are studied. For given stable polynomials \(f_n(s)\), \(g_m(s)\), of degrees \(n\), \(m\), respectively, a necessary condition for the stability of the polynomial \(\alpha f_n(s)+(1-\alpha)g_m(s)\), \(0\leq\alpha\leq 1\), is given. The question how zeros of a complex polynomial tend to infinity as some of its coefficients, the leading one included, tent to zero is investigated. Some examples are given.