Elimination of variables and the determinant of the max plus algebra (Q2880349)

The author develops the idea of the variable elimination described by \textit{N. Shinzawa} and \textit{R. Hirota} [J. Phys. A, Math. Gen. 36, No. 16, 4667--4675 (2003; Zbl 1052.37056)] to give a theoretical foundation for the determinant of the max plus algebra. A special class of max plus linear equations is considered. The max plus linear equations contain the inequality systems as a special case which correspond to the convex polyhedra. As a result, the consistency condition of the corresponding max plus linear equations is obtained. Thereafter, a new type of the determinant as a consistency condition of the class of max plus linear equations is proposed. Some examples of the determinant and the relationship between the ultradiscrete permanent are also shown.