Riemann-Roch for Sub-Lattices of the Root Lattice $A_n$

Abstract

Recently, Baker and Norine (Advances in Mathematics, 215(2): 2007) found new analogies between graphs and Riemann surfaces by developing a Riemann-Roch machinery on a finite graph G. In this paper, we develop a general Riemann-Roch Theory for sub-lattices of the root lattice An by following the work of Baker and Norine, and establish connections between the Riemann-Roch theory and the Voronoi diagrams of lattices under certain simplicial distance functions. In this way, we rediscover the work of Baker and Norine from a geometric point of view and generalise their results to other sub-lattices of An. In particular, we provide a geometric approach for the study of the Laplacian of graphs. We also discuss some problems on classification of lattices with a Riemann-Roch formula as well as some related algorithmic issues.Recently, Baker and Norine [2] proved a graph theoretic analogue of the classical Riemann-Roch theorem for curves in algebraic geometry. The proof is combinatorial and makes use of chip-firing games [5] and parking functions on graphs. Several papers later extended the results of Baker and Norine to tropical curves [15,18,21]. The question treated in this paper is to characterize those lattices which admit a Riemann-Roch theorem for the corresponding analogue of the rank-function defined by Baker and Norine.

Commutative algebra of generalised Frobenius numbers

“…, a n ) ⊥ ∩ Z n , this data is encoded in the Hilbert series of S with the Z n /L-grading. L (3,5,8) for k from 1 to 6. The Hilbert series of the polynomial ring with the (a 1 , .…”

“…We generalise the lattice module M L to the k-th lattice module M L is the S-module generated by Laurent monomials x w such that w dominates at least k lattice points. Figure 1 shows the staircase diagram corresponding to the 3rd lattice module of the lattice L (3,4,11). By construction, the first lattice module M…”

“…Let (a 1 , a 2 , a 3 ) =(3,5,8) and hence, L(3, 5, 8) = (3, 5, 8) ⊥ ∩ Z 3 . The first Frobenius number is 7.…”

“…(k) L(3,5,8) are the integers i in the interval [0, 7] such that h m k +i ≥ k. The corresponding structure posets are shown inFigure 6.…”

Chip-firing based methods in the Riemann–Roch theory of directed graphs

“…In 2007, Baker and Norine proved a graph-theoretic analogue of the classical Riemann-Roch theorem for algebraic curves [3]. This result inspired much research about Riemann-Roch theorems on tropical curves, lattices and directed graphs [1,2,10].…”