  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 $E\lapprox 1$ GeV which make use
as input, for the first time, of the most accurate experimental data available
at $E > 1$ GeV for both $\pi K\to\pi K$ and $\pi\pi\to K\bar{K}$ 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,
matching with the SU(3) chiral expansion at NLO is performed.
