Tropical Cramer Determinants Revisited

By Marianne Akian, Stephane Gaubert, Alexander E. Guterman. 2013

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We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are en- riched with an information of multiplicity, sign, or argument. We obtain exis- tence or uniqueness results, which extend or rene earlier results of Gondran and Minoux (1978), Plus (1990), Gaubert (1992), Richter-Gebert, Sturmfels and Theobald (2005) and Izhakian and Rowen (2009). Computational issues are also discussed; in particular, some of our proofs lead to Jacobi and Gauss- Seidel type algorithms to solve linear systems in suitably extended tropical semirings.

Subfields: Algebra and Number Theory, Computational Theory and Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/7a026da043dec8eeffed811388616fb460717748