Existence and Uniqueness of Homogeneous Linear Equation Solutions in Supertropical Algebra

This research investigates the existence and uniqueness of solutions to homogeneous linear equations in supertropical algebra. We analyze the structure of supertropical matrices to identify the conditions in which nontrivial solutions exist for the system of equations A⊗𝒙 ⊨ 𝜀, where A is a matrix over a supertropical semiring and x is a vector. By applying determinant-based criteria, we demonstrate how tropical and supertropical values influence the solution space. The research applies theorems that determine the presence of trivial and nontrivial solutions and uses examples to illustrate practical methods for solving homogeneous matrix systems. This highlights the distinct characteristics of supertropical algebra compared to classical linear algebra. Our findings provide a deeper insight into solution behaviors in supertropical systems, paving the way for further research in tropical mathematics.