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Tropical roots of tropical polynomials have been previously studied and usedto localize roots of classical polynomials and eigenvalues of matrixpolynomials. We extend the theory of tropical roots from tropical polynomialsto tropical Laurent series. Our proposed definition ensures that, as in thepolynomial case, there is a bijection between tropical roots and slopes of theNewton polygon associated with the tropical Laurent series. We show that,unlike in the polynomial case, there may be infinitely many tropical roots;moreover, there can be at most two tropical roots of infinite multiplicity. Wethen apply the new theory by relating the inner and outer radii of convergenceof a classical Laurent series to the behavior of the sequence of tropical rootsof its tropicalization. Finally, as a second application, we discusslocalization results both for roots of scalar functions that admit a localLaurent series expansion and for nonlinear eigenvalues of regular matrix valuedfunctions that admit a local Laurent series expansion.

Tropical Laurent series, their tropical roots, and localization results for the eigenvalues of nonlinear matrix functions