How Roy's Equations Resolve the Up-Down Ambiguity and Reproduce theS*Resonance

By making simple Regge-pole and resonance estimates of the high-energy and intermediate-energy $\pi \pi$ scattering amplitude and using recent $\pi \pi$ experimental data, we are able to show that both the rapid rise of the $I=0\mathrm{}$ $s$-wave phase shift near the $K\overline{K}$ threshold and the consequent resolution of the up-down ambiguity are consistent with the rigorous equations of Roy.