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- Title
Two-Variable Zeta-Functions on Graphs and Riemann–Roch Theorems.
- Authors
Lorenzini, Dino
- Abstract
We investigate, in this article, a generalization of the Riemann–Roch theorem for graphs of Baker and Norine, with a view toward identifying new objects for which a two-variable zeta-function can be defined. To a lattice Λ of rank n−1 in and perpendicular to a positive integer vector R, we define the notions of g-number and of canonical vector, in analogy with the notions of genus and canonical class in the theory of algebraic curves. When Λ is the full sublattice of perpendicular to R, its g-number turns out to be the classical Frobenius number of the coefficients of R. We investigate the existence of canonical vectors—in particular, in the context of arithmetical graphs—where we obtain an existence theorem using methods from arithmetic geometry. We show that a two-variable zeta-function can be defined when a canonical vector exists.
- Subjects
ZETA functions; RIEMANN-Roch theorems; MATHEMATICAL variables; CANONICAL correlation (Statistics); LATTICE dynamics; FROBENIUS algebras
- Publication
IMRN: International Mathematics Research Notices, 2012, Vol 2012, Issue 22, p5100
- ISSN
1073-7928
- Publication type
Article
- DOI
10.1093/imrn/rnr227