Mathematics and Computational Sciences

Exploring Rank, Determinant of tropical toeplitz matrices and Solvability of tropical toeplitz linear system

Articles in Press

Document Type : Original Article

Authors

Amutha B 1
Jackson J 2
Aakash M 3
Aruna S 4
Perumal R 5

1 Department of Mathematics, St. Joseph's College of Engineering, OMR, Chennai.

2 Department of Mathematics, SRM Institute of Science and Technology, Kattankulathur, 603203.

3 St.Joseph's College of Engineering, OMR, Chennai, 600119

4 Department of Mathematics, St.Joseph's, OMR,600119

5 Department of Mathematics, SRMIST, Kattankuthur, 603203

https://doi.org/10.30511/mcs.2026.2077462.1601
Abstract
Toeplitz matrices exhibit a fascinating characteristic in that their elements are constant along each diagonal. This inherent structure not only gives rise to several intriguing theoretical properties but also holds significant importance due to its strong impact on practical applications. In tropical algebra, the tropical algebraic eigenvalues, ranks, and determinants of Toeplitz matrices can be obtained without computing the coefficients of their characteristic max-polynomials, which is a notable advantage of tropical Toeplitz matrices. In recent years, cryptographic algorithms based on tropical linear and nonlinear Toeplitz systems have been shown to be vulnerable to attacks that rely on solving tropical Toeplitz linear and nonlinear systems. Our research is motivated by the close relationship between Toeplitz and Hankel matrices, as well as by the important role their determinants play in various branches of mathematics. Additionally, our research aims to explore attack techniques for tropical Toeplitz-based cryptographic protocols, which involve solving tropical linear and nonlinear systems. In this paper, we analyze the determinants and various notions of rank for different classes of tropical Toeplitz matrices. We establish conditions under which the determinant is singular or nonsingular. Furthermore, we investigate the conditions under which tropical Toeplitz matrices are balanced or unbalanced. Finally, we identify conditions for the existence and uniqueness of solutions to tropical Toeplitz linear systems.

Keywords

Tropical linear system
Tropical determinant
Tropical rank
Tropical toeplitz matrix
Subjects
Mathematics and Computational Sciences

Articles in Press, Accepted Manuscript
Available Online from 29 May 2026

  • Receive Date 11 November 2025
  • Revise Date 17 February 2026
  • Accept Date 29 May 2026

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