Low rank perturbation of regular matrix polynomials

Let A(λ) be a complex regular matrix polynomial of degree ? with g elementary divisors corresponding to the finite eigenvalue λ₀. We show that for most complex matrix polynomials B(λ) with degree at most ? satisfying rank the perturbed polynomial (A+B)(λ) has exactly elementary divisors corresponding to λ₀, and we determine their degrees. If does not exceed the number of λ₀-elementary divisors of A(λ) with degree greater than 1, then the λ₀-elementary divisors of (A+B)(λ) are the elementary divisors of A(λ) corresponding to λ₀ with smallest degree, together with rank(B(λ)-B(λ₀)) linear λ₀-elementary divisors. Otherwise, the degree of all the λ₀-elementary divisors of (A+B)(λ) is one. This behavior happens for any matrix polynomial B(λ) except those in a proper algebraic submanifold in the set of matrix polynomials of degree at most ?. If A(λ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial.