Abstract: Given an uncertain characteristic polynomial p(s, q) whose coefficients depend on a parameter vector q. A meaningful question is how large perturbation for vector q can be permitted so that the perturbed polynomial preserves the same root distribution with the nominal polynomial p(s,0). This is called the maximal perturbation bounds for invariant root distribution of polynomials. For affine linear and bilinear perturbation cases this paper gives the answers and an algorithm for the above question.