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Tropical Homotopy Continuation
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Inspired by numerical homotopy methods we propose a combinatorial homotopy algorithm for finding all isolated solutions to a tropical polynomial systems of n tropical polynomials in n variables. In particular, a tropicalisation of the numerical "regeneration" technique leads to a new method for enumerating the mixed cells of a mixed subdivision. This tropical approach shares some ideas with the recent algorithm by Malajovich. However, our algorithm has several advantages. It is memoryless, parallelisable as a tree traversal, exact and relies on symbolic perturbations. Our computational experiments show that the method is competitive and especially fast on the Katsura class of examples.
Forward citations
Cited by 2 Pith papers
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On the Computation of Newton Polytopes of Eliminants
The paper gives a mixed-subdivision algorithm for computing mixed fiber polytopes, hence Newton polytopes of eliminants, with large practical speedups over tropical implicitization.
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Engineered Complete Intersections: Algorithmic Aspects
Mixed subdivisions and tropical homotopy for matroid/engineered complete intersections yield root counts and eliminant Newton polytopes, implemented and timed on applications.
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