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Tropical Homotopy Continuation

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arxiv 1601.02818 v1 pith:KV4RDC3S submitted 2016-01-12 math.CO math.AG

Anders Nedergaard Jensen This is my paper · ORCID
classification math.COmath.AG
keywords tropicalalgorithmhomotopymethodmixednumericaladvantagesapproach
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On the Computation of Newton Polytopes of Eliminants

    cs.SC 2025-02 accept novelty 7.0 of 10

    The paper gives a mixed-subdivision algorithm for computing mixed fiber polytopes, hence Newton polytopes of eliminants, with large practical speedups over tropical implicitization.

  2. Engineered Complete Intersections: Algorithmic Aspects

    cs.SC 2026-07 accept novelty 6.5 of 10

    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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