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

Gian Maria Negri Porzio, Vanni Noferini, Leonardo Robol

TL;DR

The paper generalizes tropical roots from polynomials to tropical Laurent series, linking tropical roots to the slopes of the corresponding Newton polygon and showing that infinitely many roots may occur with at most two of infinite multiplicity. It then connects tropicalizations to analytic matrix-valued Laurent functions on annuli, proving that the endpoints of the tropical domain coincide with the radii of convergence and that eigenvalues can be localized via a Rouché-type argument using scaled tropical blocks. A practical Newton polygon framework is developed for both computation and updating under perturbations, with strategies to handle infinite index sets and to terminate updates in typical cases. The work also demonstrates concrete applications to scalar and matrix-valued problems, including adaptive contour-integral eigensolvers where tropical roots guide the choice of quadrature points and provide inclusion/exclusion regions for eigenvalues. Overall, the approach yields norm-robust, computationally cheap localization tools for nonlinear eigenvalue problems arising from Laurent series and meromorphic matrix functions, with potential impact on numerical methods for nonlinear eigenvalue problems and contour-integral solvers.

Abstract

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton 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. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.

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

TL;DR

The paper generalizes tropical roots from polynomials to tropical Laurent series, linking tropical roots to the slopes of the corresponding Newton polygon and showing that infinitely many roots may occur with at most two of infinite multiplicity. It then connects tropicalizations to analytic matrix-valued Laurent functions on annuli, proving that the endpoints of the tropical domain coincide with the radii of convergence and that eigenvalues can be localized via a Rouché-type argument using scaled tropical blocks. A practical Newton polygon framework is developed for both computation and updating under perturbations, with strategies to handle infinite index sets and to terminate updates in typical cases. The work also demonstrates concrete applications to scalar and matrix-valued problems, including adaptive contour-integral eigensolvers where tropical roots guide the choice of quadrature points and provide inclusion/exclusion regions for eigenvalues. Overall, the approach yields norm-robust, computationally cheap localization tools for nonlinear eigenvalue problems arising from Laurent series and meromorphic matrix functions, with potential impact on numerical methods for nonlinear eigenvalue problems and contour-integral solvers.

Abstract

Tropical roots of tropical polynomials have been previously studied and used to localize roots of classical polynomials and eigenvalues of matrix polynomials. We extend the theory of tropical roots from tropical polynomials to tropical Laurent series. Our proposed definition ensures that, as in the polynomial case, there is a bijection between tropical roots and slopes of the Newton 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. We then apply the new theory by relating the inner and outer radii of convergence of a classical Laurent series to the behavior of the sequence of tropical roots of its tropicalization. Finally, as a second application, we discuss localization results both for roots of scalar functions that admit a local Laurent series expansion and for nonlinear eigenvalues of regular matrix valued functions that admit a local Laurent series expansion.

Paper Structure

This paper contains 13 sections, 19 theorems, 79 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Algebra and analysis of tropical Laurent series
  3. Tropical algebra
  4. From tropical polynomials to tropical series
  5. Asymptotic behaviour of tropical roots and infinite Newton polygons
  6. Tropicalization of functions analytic on an annulus
  7. Tropical roots and radii of convergence
  8. Eigenvalue localization for Laurent series
  9. Practical computation of the Newton polygon
  10. Applications
  11. Updating the tropical roots
  12. Estimating the number of nodes in contour integrals
  13. Conclusion

Key Result

Lemma 2.3

Let $g_j(x)$ be a set of real-valued functions all defined and convex on the same interval $\Omega \subseteq \mathbb R$, and indexed over some non-empty set $I$, possibly infinite or even uncountable. Then, the largest domain of definition of the function is an interval $D \subseteq \Omega \subseteq \mathbb R$; moreover, $g(x)$ is convex on $D$.

Figures (8)

  • Figure 1: The plot of $\operatorname{\mathsf{t\!}_\times\!} f(x)$ defined in \ref{['eq:ex2']} and its tropical roots $\alpha_{j}$ in \ref{['fig:harmona']}, and a zoom in \ref{['fig:harmonZoom']}, where we can see that the nonzero $\alpha_{j}$ are the points of nondifferentiability.
  • Figure 2: Examples of possible scenarios for $g_{2}(x)$ (black dashed line). In \ref{['fig:isolated1']} the new non differentiable point is the rightmost one, hence it becomes a distinct tropical root with multiplicity one. In \ref{['fig:isolated2']} it superposes with a previous one, hence the multiplicity of the rightmost tropical root is larger than one.
  • Figure 3: The (truncated) Newton polygon of $\operatorname{\mathsf{t\!}_\times\!} f(x)$. The two tropical roots $\alpha_{\pm\infty}$ are in correspondence with the two slopes.
  • Figure 4: The (truncated) Newton polygon of $\operatorname{\mathsf{t\!}_\times\!} f(x)$. The slopes converge from below to $0$. Note that in this case the term polygon is an abuse of notation, since the convex set has an infinite number of edges.
  • Figure 5: Lines considered by the Graham Scan algorithm in Example \ref{['ex:upd-1']}. The algorithm does not terminate within a finite number of slope comparisons.
  • ...and 3 more figures

Theorems & Definitions (38)

  • Definition 2.1: Tropical Laurent series
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6: Tropical roots of Laurent series
  • Example 2.7
  • Example 2.8
  • Example 2.9
  • Example 2.10
  • ...and 28 more