Detecting tropical defects of polynomial equations

Görlach, Paul; Ren, Ya-Tao; Sommars, Jeff

doi:10.1007/s10801-019-00916-4

Cited by 4 publications

(8 citation statements)

References 31 publications

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“…This conjecture is false. It was disproved by Görlach, Ren and Sommars, using their new algorithm for tropical basis verification [14]. Here is one of the explicit examples they found.…”

Section: Realizabilitymentioning

confidence: 88%

“…Here is one of the explicit examples they found. Theorem 8.4 (Görlach et al [14]). There exist non-realizable valuated gaussoids for n = 4.…”

Section: Realizabilitymentioning

confidence: 99%

“…It rests on using state-of-the-art software from the field of SAT solvers [40,41]. The second, third, and fourth column report the number of orbits under the group actions described in (14). Theorem 4.1 for n = 3 is Example 2.1, where the 11 gaussoids are listed.…”

Section: Census Of Small Gaussoidsmentioning

confidence: 99%

“…This count was verified independently using Thurley's #SAT solver sharpSAT [40] on the same input. To group the gaussoids into orbits under the actions of the three finite groups in (14) we wrote our own code in sage [33]. The numbers of orbits we found are those in the table.…”

Section: Census Of Small Gaussoidsmentioning

confidence: 99%

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The Geometry of Gaussoids

et al. 2018

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A gaussoid is a combinatorial structure that encodes independence in probability and statistics, just like matroids encode independence in linear algebra. The gaussoid axioms of Lněnička and Matúš are equivalent to compatibility with certain quadratic relations among principal and almost-principal minors of a symmetric matrix. We develop the geometric theory of gaussoids, based on the Lagrangian Grassmannian and its symmetries. We introduce oriented gaussoids and valuated gaussoids, thus connecting to real and tropical geometry. We classify small realizable and non-realizable gaussoids. Positive gaussoids are as nice as positroids: they are all realizable via graphical models. a 12|3 maps to σ 12 σ 33 − σ 13 σ 23 whereas a 13|2 maps to −(σ 12 σ 23 − σ 13 σ 22 ). Maintaining this sign convention is important to keep the algebra consistent with its statistical interpretation.Let J n denote the ideal generated by all homogeneous polynomials in the kernel of the map above. This defines an irreducible variety V (J n ) of dimension n+1 2 in the projective space P 2 n−2 (4+( n 2 ))−1 whose coordinates are P ∪ A. There is a natural projection from LGr(n, 2n) onto V (J n ), obtained by deleting all minors that are neither principal nor almost-principal. This is analogous to [30, Observation III.12], where the focus was on principal minors p I .Proposition 1.1. The degree of the projective variety of principal and almost-principal minors coincides with the degree of the Lagrangian Grassmannian. For n ≥ 2, it equals degree(V (J n )) = degree(LGr(n, 2n)) = n+1 2 ! 1 n · 3 n−1 · 5 n−2 · · · (2n − 1).

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“…This conjecture is false. It was disproved by Görlach, Ren and Sommars, using their new algorithm for tropical basis verification [14]. Here is one of the explicit examples they found.…”

Section: Realizabilitymentioning

confidence: 88%

“…Here is one of the explicit examples they found. Theorem 8.4 (Görlach et al [14]). There exist non-realizable valuated gaussoids for n = 4.…”

Section: Realizabilitymentioning

confidence: 99%

Section: Census Of Small Gaussoidsmentioning

confidence: 99%

Section: Census Of Small Gaussoidsmentioning

confidence: 99%

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The Geometry of Gaussoids

et al. 2018

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“…Note that there is a doubly exponential upper bound on the degree of a tropical basis [22] (cf. also [26,10]). where x 1 , .…”

Section: Definition 32 ([6]mentioning

confidence: 88%

Complexity of deciding whether a tropical linear prevariety is a tropical variety

Grigoriev

Vorobjov

2019

AAECC

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We give an algorithm, with a singly exponential complexity, deciding whether a tropical linear prevariety is a tropical linear variety. The algorithm relies on a criterion to be a tropical linear variety in terms of a duality between the tropical orthogonalization A ⊥ and the double tropical orthogonalization A ⊥⊥ of a subset A of the vector space (R ∪ {∞}) n . We also give an example of a countable family of tropical hyperplanes such that their intersection is not a tropical prevariety.

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