  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.

High Energy Physics - Phenomenology