arXiv subjects

Hanieh Tavakolipour

Publications and source records attributed to Hanieh Tavakolipour.

4 recordsLinked to original sources

arXiv2026

Properties of the Tropical Characteristic Polynomial of Symmetric Matrices

We investigate the combinatorial structure of the tropical characteristic polynomial of symmetric matrices using the tropical permanents of their principal submatrices. We establish new inequalities for the leading coefficients of the tropical characteristic polynomial, revealing concavity properties of the coefficient sequence and yielding necessary conditions for a sequence to arise as the coefficient sequence of the tropical characteristic polynomial of a symmetric matrix. These results provide a deeper understanding of the structure of tropical characteristic polynomials associated with symmetric matrices.

math.CO↗

arXiv2026

Generalized Eigenvectors and Rayleigh bounds for tropical algebraic eigenvalues

In this paper, we review the eigenpair problem in the context of tropical algebra. An important fact that has been largely overlooked in spectral theory of tropical algebra is that the tropical algebraic eigenvalues, which are obtained from the characteristic polynomial, may not correspond to any tropical eigenvector satisfying the standard eigenvalue-eigenvector equation. To resolve this, we use the tropical numerical range and define a generalized tropical eigenvalue-eigenvector relation. We define any non-zero vector satisfying this equation as a generalized tropical eigenvector. We show that a generalized tropical eigenvector always exists for any given tropical algebraic eigenvalue. We propose a computationally inexpensive method for the construction of these vectors. Additionally, we prove an upper bound for the algebraic eigenvalues of a tropical matrix, using the tropical Rayleigh quotients.

math.RA↗

arXiv2025

Spectral Properties of Positive Definite Matrices over Symmetrized Tropical Algebras and Valued Ordered fields

We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the eigenvalues and eigenvectors of these matrices. We prove that the eigenvalues of a positive (semi)-definite matrix in the tropical symmetrized setting coincide with its diagonal entries. Then, we show that the images by the valuation of the eigenvalues of a positive definite matrix over a valued nonarchimedean ordered field coincide with the eigenvalues of an associated matrix in the symmetrized tropical algebra. Moreover, under a genericity condition, we characterize the images of the eigenvectors under the map keeping track both of the nonarchimedean valuation and sign, showing that they coincide with tropical eigenvectors in the symmetrized algebra. These results offer new insights into the spectral theory of matrices over tropical semirings, and provide combinatorial formulæ for log-limits of eigenvalues and eigenvectors of parametric families of real positive definite matrices.

math.RA↗

arXiv2024

Factorization of polynomials over the symmetrized tropical semiring and Descartes' rule of sign over ordered valued fields

The symmetrized tropical semiring is an extension of the tropical semifield, initially introduced to solve tropical linear systems using Cramer's rule. It is equivalent to the real tropical hyperfield, which has been used in the study of tropicalizations of semialgebraic sets. Polynomials over the symmetrized tropical semiring, and their factorizations, were considered by Quadrat. Recently, Baker and Lorscheid introduced a notion of multiplicity for the roots of univariate polynomials over hyperfields. In the special case of the hyperfield of signs, they related multiplicities with Descarte's rule of sign for real polynomials. We investigate here the factorizations of univariate polynomial functions over symmetrized tropical semirings, and relate them with the multiplicities of roots over these semirings. We deduce a Descartes' rule for "signs and valuations", which applies to polynomials over a real closed field with a convex valuation and an arbitrary (divisible) value group. We show in particular that the inequality of the Descartes' rule is tight when the value group is non-trivial. This extends to arbitrary value groups a characterization of Gunn in the rank one case, answering also to the tightness question. Our results are obtained using the framework of semiring systems introduced by Rowen, together with model theory of valued fields.

math.RA↗