  The theoretical predictions for the subtraction constants lead to a very
accurate dispersive representation of the pi pi scattering amplitude below 0.8
GeV. The extension of this representation up to the maximum energy of validity
of the Roy equations (1.15 GeV) requires a more precise input at high energies.
In this paper we determine the trajectories and residues of the leading Regge
contributions to the pi pi amplitude (Pomeron, f and rho), using factorization,
phenomenological parametrizations of the pi N and NN total cross sections at
high energy and a set of sum rules which connect the high and low energy
properties of pi pi scattering. We find that nonleading Regge terms are
necessary in order to achieve a smooth transition from the partial waves to the
Regge representation at or below 2 GeV. We obtain thus a Regge representation
consistent both with the experimental information at high energies and the Roy
equations for the partial waves with l =< 4. The uncertainties in our result
for the Regge parameters are sizable but in the solutions of the Roy equations,
these only manifest themselves above K-Kbar threshold.
