Semigroups, graphs and applications
Coordinator: José Ignacio Farrán Martín (Universidad de Valladolid)
- Antonio Campillo López (Universidad de Valladolid)
- Félix Delgado de la Mata (Universidad de Valladolid)
- Philippe Gim\'enez (Universidad de Valladolid)
- Carlos Marijuán López (Universidad de Valladolid)
- Edgar Martínez Moro (Universidad de Valladolid)
- Carlos Munuera Gómez (Universidad de Valladolid)
- Diego Ruano Benito (Universidad de Valladolid)
- Fernando Eduardo Torres Orihuela (Universidad de Campinas, Brasil)
This interdisciplinary research group has a common interest in the theory of semigroups from various points of view. Thus, the semigroups have been considered as an interesting tool in singularity theory, in commutative algebra, in the theory of error-correcting codes, or even in graph theory, from both a theoretical and a computational viewpoint. As a result of this work, relevant papers have appeared in the field of numerical and affine semigroups, and besides the computer algebra system SINGULAR includes the (unique) library that implements algebraic geometry codes and computes Weierstrass semigroups of plane curves in positive characteristic. In a future work, some of the purposes are the computation of generalized Feng-Rao distances of numerical semigroups with their applications in coding theory and cryptography, the implementation of error-correcting codes with the aid of Graver bases, the construction of codes from graphs, and the computation of Zeta functions of graphs.
Error-correcting codes allow us to detect and correct errors that are produced in the transmission or storage of information, as in the case of CD or DVD devices. The so-called AG codes (AG stands for Algebraic Geometry) have an excellent behaviour in their parameters (dimension and minimum distance) as their length becomes arbitrarily large. In these codes, constructed from smooth algebraic curves, the computation of these parameters is closely connected to the involved Weierstrass semigroups, which are a particular case of numerical semigroups, and more precisely to the Feng-Rao distances and Feng-Rao numbers. On the other hand, the numerical semigroups are related to graphs and patterns, and part of this research group is devoted to this task since a long time ago. In particular, it would be nice to explore the possibility of generalizing the construction of AG codes, according to recent works about the Reiemann-Roch theory on graphs, and the generalization of the corresponding Weierstrass semigroups as a consequence. Finally, the study of Zeta functions and Poincaré series on graphs would be developed in a similar way as in the case of AG codes, though this task has an intrinsic interest by itself.