  We show that the Davenport-Stothers inequality from characteristic 0 fails in
any characteristic p>3. The proof uses elliptic surfaces over the projective
line and inseparable base change. We then present adjusted inequalities. These
follow from results of Pesenti-Szpiro. For characteristic 2 and 3, we achieve a
similar result in terms of the maximal singular fibres of elliptic surfaces
over the projective line. Our ideas are also related to supersingular surfaces
(in Shioda's sense).
