Generic Asymptotics of Eigenvalues and Min-Plus Algebra
Résumé
We consider a square matrix $\mathcal{A}_{\epsilon}$ whose entries have asymptotics of the form $(\mathcal{A}_{\epsilon})_{ij}= a_{ij}\epsilon^{A_{ij}}+o(\epsilon^{A_{ij}})$ when $\epsilon$ 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 $\mathcal{A}_\epsilon$. We show that the sequence of exponents of the eigenvalues of $\mathcal{A}_\epsilon$ 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 $\mathcal{A}_\epsilon$ can be computed generically by Schur complement formulæ, which extend the perturbation formulae of Vivsik, Ljusternik and Lidskii, and have fewer singular cases.
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