Automorphism groups on tropical curves: some cohomology calculations

Joyner, David; Ksir, Amy; Melles, Caroline Grant

doi:10.1007/s13366-011-0049-3

Cited by 2 publications

(1 citation statement)

References 12 publications

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“…We decide between (25) and (26) by computing the coordinates of w in two different ways: beginning from 1 and beginning from 2.…”

Section: Lemma 12mentioning

confidence: 99%

Distances on the tropical line determined by two points

Puente

2014

Kybernetika

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Let p ′ , q ′ ∈ R n . Write p ′ ∼ q ′ if p ′ − q ′ is a multiple of (1, . . . , 1). Two different points p and q in R n / ∼ uniquely determine a tropical line L(p, q), passing through them, and stable under small perturbations. This line is a balanced unrooted semi-labeled tree on n leaves. It is also a metric graph.If some representatives p ′ and q ′ of p and q are the first and second columns of some real normal idempotent order n matrix A, we prove that the tree L(p, q) is described by a matrix F , easily obtained from A. We also prove that L(p, q) is caterpillar. We prove that every vertex in L(p, q) belongs to the tropical linear segment joining p and q. A vertex, denoted pq, closest (w.r.t tropical distance) to p exists in L(p, q). Same for q. The distances between pairs of adjacent vertices in L(p, q) and the distances d(p, pq), d(qp, q) and d(p, q) are certain entries of the matrix |F |. In addition, if p and q are generic, then the tree L(p, q) is trivalent. The entries of F are differences (i.e., sum of principal diagonal minus sum of secondary diagonal) of order 2 minors of the first two columns of A.

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“…We decide between (25) and (26) by computing the coordinates of w in two different ways: beginning from 1 and beginning from 2.…”

Section: Lemma 12mentioning

confidence: 99%

Distances on the tropical line determined by two points

Puente

2014

Kybernetika

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From Polygons to Ultradiscrete Painlevé Equations

Ormerod

Yamada

2015

SIGMA

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Abstract. The rays of tropical genus one curves are constrained in a way that defines a bounded polygon. When we relax this constraint, the resulting curves do not close, giving rise to a system of spiraling polygons. The piecewise linear transformations that preserve the forms of those rays form tropical rational presentations of groups of affine Weyl type. We present a selection of spiraling polygons with three to eleven sides whose groups of piecewise linear transformations coincide with the Bäcklund transformations and the evolution equations for the ultradiscrete Painlevé equations.

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Automorphism groups on tropical curves: some cohomology calculations

Cited by 2 publications

References 12 publications

Distances on the tropical line determined by two points

Distances on the tropical line determined by two points

From Polygons to Ultradiscrete Painlevé Equations

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