The role of the input in Roy's equations for pion-pion scattering

The Roy equations determine the S- and P-wave pion-pion phase shifts on a low energy interval. They allow the derivation of threshold parameters from experimental data. We examine the solutions of these equations that are in the neighborhood of a given solution by means of a linearization procedure. An updated survey of known results on the dimension of the manifold of solutions is presented. The solution is unique for a low energy interval with upper end at 800 MeV. We determine its response to small variations of the input: S-wave scattering lengths and absorptive parts above 800 MeV. We confirm the existence of a universal curve of solutions in the plane of the S-wave scattering lengths and provide a control of the decrease of the influence of the input absorptive parts with increasing energy. A general result on the suppression of unphysical singularities at the upper end of the low energy interval is established and its practical implications are discussed.