Symmetries of explicit Runge-Kutta methods (Q2454393)

This paper refines the abstract algebraic approach recently presented by the author to compute the solution of the order conditions for Runge-Kutta methods. Actually, such conditions are a system of a large number of polynomial equations in a large number of unknowns. The abstract algebraic approach is based on abstract algebra properties of varieties of matrices defined by the order equations. Here this approach is improved by introducing the one-dimensional symmetries, which allow to reduce the list of unknowns. The main result of this paper proves that the first three columns of the Butcher tableau can be expressed in terms of the others and that one unknown can be excluded.