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arxiv: 1309.6298 · v1 · submitted 2013-09-24 · 🧮 math.CO · math.OC

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Tropical Cramer Determinants Revisited

Marianne Akian , St\'ephane Gaubert , Alexander Guterman
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classification 🧮 math.CO math.OC
keywords tropicalcramerlinearresultssystemstypealgorithmsargument
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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 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.

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  1. Solving one-sided linear systems over symmetrized and supertropical semiring

    math.RA 2026-05 unverdicted novelty 5.0

    The authors extend the polynomial-time greatest solution approach and the search for minimal solutions for linear systems to symmetrized and supertropical semirings, with implications for tropical cryptography.