A new analysis of

With the aim of generating new constraints on the OZI suppressed couplings of chiral perturbation theory a set of six equations of the Roy and Steiner type for the S- and P-waves of the $\pi K$ scattering amplitudes is derived. The range of validity and the multiplicity of the solutions are discussed. Precise numerical solutions are obtained in the range $E\lesssim 1$ GeV which make use as input, for the first time, of the most accurate experimental data available at $E\gtrsim 1$ GeV for both $\pi K\to\pi K$ and $\pi\pi\to K\overline{K}$ amplitudes. Our main result is the determination of a narrow allowed region for the two S-wave scattering lengths. Present experimental data below 1 GeV are found to be in generally poor agreement with our results. A set of threshold expansion parameters, as well as sub-threshold parameters are computed. For the latter, a matching with the SU(3) chiral expansion at NLO is performed