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Book Sections Year : 2014

Tropical Cramer Determinants Revisited

Abstract

We prove general Cramer type theorems for linear systems over various extensions of the tropical semiring, in which tropical numbers are enriched with an information of multiplicity, sign, or argument. We obtain existence or uniqueness results, which extend or refine 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.

Dates and versions

hal-00881203 , version 1 (07-11-2013)

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Marianne Akian, Stéphane Gaubert, Alexander Guterman. Tropical Cramer Determinants Revisited. G.L. Litvinov and S.N. Sergeev. Tropical and Idempotent Mathematics and Applications, 616, AMS, pp.45, 2014, Contemporary Mathematics. ⟨hal-00881203⟩
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