Riemann–Roch and Abel–Jacobi theory on a finite graph

Baker, Matthew; Norine, Serguei

doi:10.1016/j.aim.2007.04.012

Cited by 372 publications

(708 citation statements)

References 23 publications

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“…As alluded to in [BN,Remark 4.8], one can use Lemma 5.7 to obtain an alternate proof of Corollary 4.7 from [BN] which does not make use of circuit theory.…”

Section: 2mentioning

confidence: 99%

Harmonic Morphisms and Hyperelliptic Graphs

Baker

Norine

2009

International Mathematics Research Notices

166

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Abstract. We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical RiemannHurwitz formula, study the functorial maps on Jacobians and harmonic 1-forms induced by a harmonic morphism, and present a discrete analogue of the canonical map from a Riemann surface to projective space. We also discuss several equivalent formulations of the notion of a hyperelliptic graph, all motivated by the classical theory of Riemann surfaces. As an application of our results, we show that for a 2-edge-connected graph G which is not a cycle, there is at most one involution ι on G for which the quotient G/ι is a tree. We also show that the number of spanning trees in a graph G is even if and only if G admits a non-constant harmonic morphism to the graph B2 consisting of 2 vertices connected by 2 edges. Finally, we use the Riemann-Hurwitz formula and our results on hyperelliptic graphs to classify all hyperelliptic graphs having no Weierstrass points.

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“…As alluded to in [BN,Remark 4.8], one can use Lemma 5.7 to obtain an alternate proof of Corollary 4.7 from [BN] which does not make use of circuit theory.…”

Section: 2mentioning

confidence: 99%

Harmonic Morphisms and Hyperelliptic Graphs

Baker

Norine

2009

International Mathematics Research Notices

166

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“…The pBBS is obtained from the box-ball system (BBS), which was introduced by Takahashi and Satsuma as a soliton cellular automaton [33], by imposing a periodic boundary condition. The procedure which reduces (2) to (28) is called the ultradiscretization [35]. Now we introduce the spectral curve of the UD-pTL by using the procedure of ultradiscretization.…”

Section: Ud-ptl and Its Spectral Curvementioning

confidence: 99%

“…We denote it by rank L(D). The following theorem (a corollary of the tropical Riemann-Roch theorem) is useful [5,25,2].…”

Section: Addition On Tropical Hyperelliptic Curvesmentioning

confidence: 99%

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A geometric realization of the periodic discrete Toda lattice and its tropicalization

Nobe¹

2013

J. Phys. A: Math. Theor.

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An explicit formula concerning curve intersections equivalent to the time evolution of the periodic discrete Toda lattice is presented. First, the time evolution is realized as a point addition on a hyperelliptic curve, which is the spectral curve of the periodic discrete Toda lattice, then the point addition is translated into curve intersections. Next, it is shown that the curves which appear in the curve intersections are explicitly given by using the conserved quantities of the periodic discrete Toda lattice. Finally, the formulation is lifted to the framework of tropical geometry, and a tropical geometric realization of the periodic box-ball system is constructed via tropical curve intersections.

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“…• In [3], a version of the Riemann-Roch theorem for graphs was proved by purely combinatorial methods. Shortly afterwards Gathmann and Kerber [7] used the result to prove Riemann-Roch theorem for tropical curves.…”

Section: Introductionmentioning

confidence: 99%

Rank of divisors on tropical curves

Hladký

Kráľ

Norine

2013

Journal of Combinatorial Theory, Series A

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We investigate, using purely combinatorial methods, structural and algorithmic properties of linear equivalence classes of divisors on tropical curves. In particular, we confirm a conjecture of Baker asserting that the rank of a divisor D on a (non-metric) graph is equal to the rank of D on the corresponding metric graph, and construct an algorithm for computing the rank of a divisor on a tropical curve.

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Riemann–Roch and Abel–Jacobi theory on a finite graph

Cited by 372 publications

References 23 publications

Harmonic Morphisms and Hyperelliptic Graphs

Harmonic Morphisms and Hyperelliptic Graphs

A geometric realization of the periodic discrete Toda lattice and its tropicalization

Rank of divisors on tropical curves

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