# Analysis of linear systems over idempotent semifields

## TL;DR

This paper explores methods for solving linear systems over idempotent semifields, comparing their approaches and showing that various methods yield the same maximal solutions.

## Contribution

It introduces and analyzes multiple solution methods, including pseudo-inverse, Cramer's rule, and normalization, highlighting their equivalence in maximal solutions.

## Key findings

- All methods produce identical maximal solutions.
- The normalization method explicitly involves the constant vector.
- The pseudo-inverse and Cramer's rule relate to system solvability.

## Abstract

In this paper, we present and analyze methods for solving a system of linear equations over idempotent semifields. The first method is based on the pseudo-inverse of the system matrix. We then present a specific version of Cramer's rule which is also related to the pseudo-inverse of the system matrix. In these two methods, the constant vector plays an implicit role in solvability of the system. Another method is called the normalization method in which both the system matrix and the constant vector play explicit roles in the solution process. Each of these methods yields the maximal solution if it exists. Finally, we show the maximal solutions obtained from these methods and some previous methods are all identical.