On Davenport-Stothers inequalities and elliptic surfaces in positive   characteristic
  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). 
Matthias Schuett, Andreas Schweizer