ar X iv : m at h / 04 02 09 0 v 1 [ m at h . SP ] 5 F eb 2 00 4 GENERIC ASYMPTOTICS OF EIGENVALUES AND MIN - PLUS ALGEBRA

We consider a square matrix Aǫ whose entries have asymptotics of the form (Aǫ) ij = a ij ǫ A ij + o(ǫ A ij) when ǫ goes to 0, for some complex coefficients a ij and real exponents A ij. We look for asymptotics of the same type for the eigenvalues of Aǫ. We show that the sequence of exponents of the eigenvalues of Aǫ is weakly (super) majorized by the sequence of corners of the min-plus characteristic polynomial of the matrix A = (A ij), and that the equality holds for generic values of the coefficients a ij. We derive this result from a variant of the Newton-Puiseux theorem which applies to asymptotics of the preceding type. We also introduce a sequence of generalized minimal circuit means of A, and show that this sequence weakly majorizes the sequence of corners of the min-plus characteristic polynomial of A. We characterize the equality case in terms of perfect matching. When the equality holds, we show that the coefficients of all the eigenvalues of Aǫ can be computed generically by Schur complement formulae, which extend the perturbation formulae of Višik, Ljusternik and Lidski˘ ı, and have fewer singular cases.