Gröbner bases for mesh relations and applications to compositions of irreducible morphisms

We give a necessary condition for the existence of a path of n irreducible morphisms between indecomposable modules whose composition lies in the (n + 1)-power of the radical. In order to do that, we consider the general criterion given by C. Chaio, P. Le Meur and S. Trepode, which relates these compositions with zero paths in the mesh category, and then study morphisms in the mesh category by providing Gröbner bases for the subspaces generated by the mesh relations.