Stringent skew hyperfields are perfect: over them every vector of a matroid is orthogonal to every covector, and weak matroids coincide with strong matroids.
Real Tropical Singularities and Bergman Fans
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abstract
In this paper, we classify singular real plane tropical curves by means of subdivisions of Newton polytopes. First, we introduce signed Bergman fans (generalizing positive Bergman fans from [AKW06]) that describe real tropicalizations of real linear spaces ([Tab15]). Then, we establish a duality of real plane tropical curves and signed regular subdivisions of the Newton polytope and explore the combinatorics. We define a signed secondary fan that parametrizes real tropical Laurent polynomials and study the subset providing singular real plane tropical curves. A cone of the signed secondary fan is of maximal dimensional type if its corresponding subdivision contains only marked points ([MMS12a]). These cones parametrize real plane tropical curves. We classify singular real plane tropical curves of maximal dimensional type.
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math.CO 1years
2019 1verdicts
CONDITIONAL 1representative citing papers
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Perfect matroids over hyperfields
Stringent skew hyperfields are perfect: over them every vector of a matroid is orthogonal to every covector, and weak matroids coincide with strong matroids.