Some properties of zeros of polynomials with vanishing coefficients

It is well-known that when a polynomial whose coefficients are continuous functions of a parameter loses its degree then some of its zeros must vanish at infinity. In this paper, we consider such a situation: we examine how roots of a complex polynomial tend to infinity as some of its coefficients, including the leading one, tend to zero. We show, among other things, that in such a situation the unbounded paths traced by the roots of the polynomial have asymptotes; we also obtain their formulas. Some examples are presented to complete and illustrate the results.