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Elimination of variables and the determinant of the max plus algebra. (English) Zbl 1248.15024
The author develops the idea of the variable elimination described by N. Shinzawa and 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.

MSC:
15A80 Max-plus and related algebras
15A15 Determinants, permanents, traces, other special matrix functions
15A06 Linear equations (linear algebraic aspects)
15A45 Miscellaneous inequalities involving matrices
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