Matroids over hyperfields

Baker, Matthew J.; Bowler, Nathan

doi:10.48550/arxiv.1601.01204

Cited by 17 publications

(57 citation statements)

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“…Baker and Bowler's framework of matroids over hyperfields, and more generally, matroids over tracts, has provided a more general framework for tropical Plücker vectors and tropical linear spaces [2]. A totally ordered idempotent semifield S defines a hyperaddition operator ⊞ by…”

Section: Preliminariesmentioning

confidence: 99%

“…Replacing addition with hyperaddition makes S a hyperfield, and matroids over this hyperfield in the sense of [2] are equivalent to tropical linear spaces; in terms of hyperfields, L w is the set of covectors in the sense of [1].…”

Section: Preliminariesmentioning

confidence: 99%

“…Frenk defined the minors of a tropical linear space in [7], following the definition of the minors of a valuated matroid given by Dress and Wenzel [5]. Minors of matroids over hyperfields were defined by Baker and Bowler [2]. The following is essentially [2, Lemma 6.4], translated to the language of totally ordered idempotent semifields.…”

Section: Matroidsmentioning

confidence: 99%

“…A tropical linear space is realizable with respect to a particular valuation if it is a tropicalization of some linear space under that valuation. For matroids, a realizable tropical linear space via k → B is exactly a representable matroid over k. From the perspective of matroids over hyperfields, a valuation val : k → S induces a hyperfield homomorphism from k to the canonical hyperfield associated to S, and tropicalization is exactly the pushforward under this hyperfield homomorphism [2].…”

Section: 2mentioning

confidence: 99%

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The image of a tropical linear space

Mundinger

2018

Preprint

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Given a tropical linear space L ⊆ T n and a matrix A ∈ T m×n , the image AL of L under A is typically not a tropical linear space. We introduce a tropical linear space tropim A (L), the tropical image, containing AL. We show under mild hypotheses that tropim A (L) is realizable if L is and apply the tropical image to construct the stable sum of two tropical linear spaces without a disjoint pair of bases.

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Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

Section: Matroidsmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

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The image of a tropical linear space

Mundinger

2018

Preprint

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“…They give a phased analog to the chirotope and circuit axioms of oriented matroids, and show that the definitions are cryptomorphic. A recent paper of Baker and Bowler [2], yet to appear as of the writing of this introduction, generalizes oriented matroid and phased matroid properties into matroids over hyperfields, proving the definitions are cryptomorphic on the hyperfield level, and uniting the fields of oriented matroids, phased matroids, and matroids over hyperfields other than signs of real numbers and phases of complex number.…”

Section: Introductionmentioning

confidence: 99%

Realization Spaces of Uniform Phased Matroids

Ruiz

2018

Preprint

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A phased matroid is a matroid with additional structure which plays the same role for complex vector arrangements that oriented matroids play for real vector arrangements.The realization space of an oriented (resp., phased) matroid is the space of vector arrangements in R n (resp., C n ) that correspond to oriented (resp., phased) matroid, modulo a change of coordinates. According to Mnëv's Universality Theorem, the realization spaces of uniform oriented matroids with rank greater than or equal to 3 can be as complicated as any open semi-algebraic variety.In contrast, uniform phased matroids which are not essentially oriented have remarkably simple realization spaces if they are uniform.We also present a criterion for realizability of uniform phased matroids that are not essentially oriented.

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Tropical Spectrahedra

Allamigeon

Gaubert

Skomra

2020

Discrete Comput Geom

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We introduce tropical spectrahedra, defined as the images by the nonarchimedean valuation of spectrahedra over the field of real Puiseux series. We provide an explicit polyhedral characterization of generic tropical spectrahedra, involving principal tropical minors of size at most 2. One of the key ingredients is Denef-Pas quantifier elimination result over valued fields. We obtain from this that the nonarchimedean valuation maps semialgebraic sets to semilinear sets that are closed. We also prove that, under a regularity assumption, the image by the valuation of a basic semialgebraic set is obtained by tropicalizing the inequalities that define it.

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Matroids over hyperfields

Cited by 17 publications

References 0 publications

The image of a tropical linear space

The image of a tropical linear space

Realization Spaces of Uniform Phased Matroids

Tropical Spectrahedra

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