Low rank perturbation of regular matrix polynomials

Let A(λ) be a complex regular matrix polynomial of degree ℓ with g elementary divisors corresponding to the finite eigenvalue λ0. We show that for most complex matrix polynomials B(λ) with degree at most ℓ 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 ℓ. If A(λ) has an infinite eigenvalue, the corresponding result follows from considering the zero eigenvalue of the perturbed dual polynomial.