  We discuss the existence of the light scalar meson K^*_0(800) (also called
kappa) in a rigorous way, by showing the presence of a pole in the pi K --> pi
K amplitude on the second Riemann sheet. For this purpose, we study the domain
of validity of two classes of Roy-Steiner representations in the complex energy
plane. We prove that one of them is valid in a region sufficiently broad in the
imaginary direction. From this representation, we compute the l=0 partial wave
in the complex plane with neither additional approximation nor model
dependence, relying only on experimental data. A scalar resonance with
strangeness S=1 is found with the following mass and width: E_kappa = 658 \pm
13 MeV and Gamma_kappa = 557 \pm 24 MeV.
